61FT  OP 
ROBERT 
BE1PHBR. 


PRACTICAL 


HYDRAULICS 


BY 

P.  M.  KANDALL, 

AUTHOR  OF 

QUARTZ    OPERATORS'    HAND-BOOK." 


PUBLISHED  AND  SOLD  BY 

Y   &   Oo. 

PROPRIETORS    MINING    AND    SCIENTIFIC    PRESS, 
SAN    FRANCISCO,  CAL.,   1886. 


Entered  according  to  the  Act  of  Congress,  in  the  Librarian's  Office, 
Washington,  D.  C.,  1886,  by  DBWEY  &  Co. 


PREFACE. 


The  present  work  is  designed  as  a  true  exposition  01  the  prin- 
ciples and  application  of  those  branches  of  hydraulics,  of  which  it 
treats. 

The  necessity  of  such  a  work  at  this  time  will  be  obvious  to 
those  who  shall  have  compared  the  results  deduced  from  the  for- 
mulas of  nearly  all  our  most  noted  authors  on  hydraulics,  with  the 
results  of  observation.  Thus,  the  formulas  of  DuBuat,  Eytelwein, 
Girard,  Prony,  D'Aubuisson,  Neville,  Leslie,  Pole,  Beardmore  and 
Hagen,  enjoying  the  reputation  of  standard  authors,  give  as  by  the 
data  at  hand,  with  respect  to  the  flow  of  water  in  open  streams  of 
medium  size,  results  varying  from  fifteen  to  one  hundred  and 
twenty-five  per  cent  in  excess  ot  the  observed  results,  and  in 
large  streams,  results  varying  from  thirty  to  sixty-seven  per  cent 
below  those  observed.  These  errors  are  radical.  The  defect! venesa 
of  these  and  other  works  heretofore  regarded  standard  on  this 
highly  important  branch  of  hydraulics,  is  well  portrayed  by  the 
following  extracts  from  an  article  in  an  Engliah  periodical,  "Engi- 
neering" of  Dec.  31,  1875,  entitled  "Hydraulic  Experiments,"  viz  : 
"  The  tabulated  velocities  (in  Neville's  work,  based  upon  DuBuat) 
"  though  expressed  in  hundredths  of  an  inch,  are  in  reality  but  the 
"  wildest  guesses  at  the  actual  velocities  in  irrigation  canals  of 
"  ordinary  dimensions.  Colonel  Cautley  relied  upon  DuBuat  when 
"  he  laid  out  the  Ganges  Canal,  and  found  him  but  a  rotten  reed, 
"  for  the  water  in  every  instance  tore  along  at  an  unexpected  velo- 
"  city,  and  the  erosion  of  the  bed  and  destruction  of  the  works 
"  followed  in  its  wake.  Da  Buat  then  must  be  put  upon  the  top 
"  shelf  of  the  book-case,  and  it  will  be  just  as  well  when  the  steps 

164453 


IV  PKEFACE. 

"  are  there  to  carry  up  every  English  work,  in  which  the  names  of 
"  Branning,  Girard,  Bossut,  Prony,  Eytelwein,  or  D'Aubuisson  are 
"  continually  recurring  as  authorities  against  whom  no  action  can 
"  be  taken.  In  this  general  clearance  Baardmore,  Downing,  Box, 
"  and  almost  every  other  hydraulic  text  book  compiled  by  English- 
"  men,  will  with  more  or  less  hesitation  have  been  shelved." 

Again,  "  in  1880,"  says  L.  D.  A.  Jackson,  in  his  Hydraulic  Man- 
ual "  the  extensive  experiments  of  Captain  Allan  Cunningham  on 
"  the  Ganges  Canal,  have  substantiated  the  truth  of  Kutter's  laws 
"  when  applied  to  very  large  canals,  and  dealt  the  final  blow  to  the 
"  velocity  formulas  of  all  the  older  hydraulicians." 

In  the  main  text  of  the  present  work  it  is  stated  that  D'Arcy 
in  1835,  and  Bazin,  in  1865,  published  formulas  better  adapted  than 
any  preceding  for  finding  the  flow  of  water  in  open  streams  and 
pipes  of  medium  size;  that  Humphreys  and  Abbot  published  in 
1861,  formulas  suited  to  the  determination  of  flow  of  large  streams, 
but  not  to  the  flow  of  small  streams,  and  that  the  wide  gap 
between  the  formulas  of  D'Arcy  and  Bazin,  and  those  <5f 
Humphreys  and  Abbot  were  effectually  closed  up  in  1870  by 
the  introduction  of  Kutter's  formula.  We  will  now  add,  that 
this  achievement  with  respect  to  hydraulic  science  seems  to  us 
the  masterpiece  of  the  nineteenth  century.  The  Kutter  formula 
applies  equally  well  to  small,  medium  sized  and  large  svreams. 
Farther  experiments  may  perchance  require  it  to  be  somewhat  mod- 
ified; but  so  far  as  known  at  present,  of  all  the  formulas  deduced 
for  like  purposes,  it  seems  the  nearest  approximate  to  perfection. 
The  principal  tables  computed  by  Herr  Kutter,  from  his  for- 
mula under  consideration,  give  the  coefficients  of  velocity  in  terms 
of  metrical  measures,  thereby  rendering  their  application  a  labor- 
ious task  in  the  determination  of  the  velocities  themselves  in  terms 
of  feet  measures. 

To  obviate  this  task,  Table  27  in  the  present  work  has  been 
computed  from  the  same  formula  (Kutter's)  giving  in  terms  of  feet 
measures,  the  velocities  of  flow  in  open  streams,  differing  in  regime 
and  in  slope,  and  varying  from  the  size  of  a  small  ditch  to  that  of 
the  Mississippi  River.  The  table  is  nominally  for  open  streams, 
but  is  equally  well  adapted  for  determining  the  flow  of  water  in 
pipes.  Table  17  computed  for  the  flow  in  pipes  only,  will,  for  this 


PREFACE.  V 

purpose,  be  found,  however,  still  more  convenient.  For  this  table 
we  are  indebted  in  part  to  J.  T.  Fanning's  very  admirable  treatise  on 
"Water  Supply  Engineering,"  which  indebtedness  we  hereby  re- 
spectfully acknowledge.  It  will  be  noted  however  that  we  have 
not  only  considerably  enlarged  the  original  table  of  Mr.  Fanning, 
but,  among  other  things,  conferred  upon  it  a  new  and  valuable 
feature — that  of  giving  the  quantity  of  flow  in  addition  to  the  veloc- 
ity. Each  result  set  down  in  Tables  17  and  27,  represents  essen- 
tially a  mean  of  numerous  observed  results:  hence  must  necessarily 
coincide  in  practice  with  other  results  obtained  under  like  con- 
ditions. With  respect  to  accuracy,  scope  of  application  and  ease  of 
reference,  these  tables  seem  to  meet  more  fully  the  requirements  of 
all  concerned  in  this  branch  of  hydraulic  engineering,  than  any 
others  designed  for  similar  purposes. 

Tables  28  and  29  will  be  found  very  important  auxiliaries  to 
Table  27,  in  the  ready  determination  of  the  flow  of  water  in  beds 
of  different  forms. 

Tables,  two  relating  to  the  flow  of  water  in  rectangular  weirs, 
four  to  quadrant  weirs,  seven  to  the  flow  through  rectangular  ori- 
fices, and  eight  to  the  different  values  of  the  so-called  "  miner's  inch," 
will  also  be  found  of  no  little  value  in  practice.  The  simplicity  of  the 
quadrant  weir,  its  cheapness  and  the  assurances  by  Prof.  Thompson 
of  its  superiority  over  those  of  different  forms,  induced  the  author 
of  the  work  in  hand  to  compute  Table  4. 

This  form  seems  peculiarly  well  adapted  to  the  measurement  of 
the  flow  of  small  quantities  of  water;  for  example,  from  two  to  20 
miner's  inches.  This  table,  however,  greatly  exceeds  these  limits. 
The  discussion  of  the  subjects  of  "  maximum  work  effected  by 
water  on  issuing  under  pressure  from  pipes,"  and  of  "minimum 
weight  and  consequent  minimum  cost  of  an  inverted  siphon,"  is,  so 
far  as  the  author  is  informed,  new.  By  the  application  of  the  prin- 
ciples here  demonstrated,  the  greatest  economy,  the  only  proper 
limit  or  standard  of  the  truly  practical,  is  attained. 

The  simple  plan,  pursued  in  the  preparation  of  the  present 
work  consists  :— 

1st.  In  demonstrating  concisely  the  principle,  or  principles, 
involved  in  the  subject  matter,  yet  in  a  manner  sufficiently  ample 
and  clear  to  be  readily  followed  by  the  student,  or  by  the  practi- 


VI  PREFACE. 

tioner  desiring  to  refresh  his  mind,  or  to  assure  himself  of  the  cor- 
rectness of  the  results. 

2nd.  In  expressing  in  words  the  simplest  rule  or  rules  cor- 
responding to  the  formula  or  result  of  such  demonstration. 

3d.  In  applying  the  rule  or  rules  so  derived,  to  practical  ex- 
amples with  full  and  clear  explanations;  or  in  applying  the  formula 
direct  to  the  examples,  when  it  is  too  complex  to  be  well  expressed 
in  words. 

4th.  In  providing  tables,  so  far  as  feasible,  to  meet  the  re- 
quirements of  practice. 

By  means  of  these  tables  and  the  simple  rules  given  therewith, 
most  of  the  problems  likely  to  occur  with  respect  to  the  measure- 
ment  of  water  in  motion,  as  through  vertical  openings,  over  weirs, 
in  pipes,  in  open  streams  and  through  nozzles;  with  respect  to  the 
quantity  of  water  required  for  various  mining  purposes,  and  for  the 
purposes  of  irrigation;  and  with  respect  to  the  power  of  water  as  a 
motor,  are  answered  direct,  or  readily  solved  by  anyone  familiar 
with  common  arithmetic  only,  as  well  as  by  the  skilled  engineer. 

P.  M.  RANDALL. 
San  Francisco,  March  17th,  1886. 


INDEX. 


Formulas,  and  Formulas  and  Rules  corresponding  ;  '*  F"  repre- 
senting formula,  and  "R"  rule  ;  page  referring  to  rule  : 

F.          R.  PAOB. 

(1) —Acceleration  of  gravity  at  sea  level  in  lat.  45" 2 

(3) —Acceleration  of  gravity,  the  latitude,  elevation  and  acceleration  of 

gravity  at  lat.  45*.  given 2 

(4) —Acceleration  of  gravity,  the  latitude,  force  of  gravity  at  sea  level  and 

elevation,  given 3 

(175)(176).40-Additional  head  for  angular  bend 162 

(177)  (178). 41-Additional  head  for  curved  bend 163 

(94)  (95).  .28 -Coefficients  of  discharge ;  of  partial  contraction 87 

(98) 29 -Coefficient  of  contraction P9 

(105) 30  -Coefficients  of  discharge  under  a  head  of  water  in  motion 100 

31-Coefficients  for  a  short  tube 103 

(107) 32— Discharge  through  cylindrical  tube  -  length  to  diameter  3:1 104 

(157) 35— Discharge  through  clean  pipes 155 

(160)  38  -  Diameter  due  head,  discharge  and  the  length  of  a  pipe 155 

39  -  Discharge,  head,  length  and  diameter  of  pipe ;  cases  1,  2,  3  and  4,  by 

Table  17 155 

54— Discharge  of  a  flume  whose  hydraulic  mean  depth  equals  that  of  a 

given  pipe ...200 

55— Discharge  of  a  flume  by  another  method 201 

(74) 23— Flow  through  a  parabolic  weir  whose  apex  ia  level  with  still  water. . .  52 

(284) —Formula  for  finding  slope  of  bank 218 

(285) —Formula  for  finding  wet  perimeter 218 

(286) —Formula  for  finding  mean  velocity  from  central  surface  velocity 239 

,287) —Formula  for  finding  flood-flow  of  streams 248 

(8) 1— Head,  due  velocity  and  gravity 6 

(11) 2— Head,  due  time  and  gravity 7 

(158) 36-Head  due  dimensions  and  discharge  of  a  pipe 155 

(198) 46-Head,  such  that  the  weight  of  an  inverted  siphon  shall  be  a  minimum  .180 

(101) —Imperfect  contraction  (circular) 93 

(102) —Imperfect  contraction  (rectangular) « 93 

(148) 34-Inlet  head  due  velocity 150 

(280) — Kutter's  Formula ...210 


Vlll  ,  •  INDEX. 

P.  R.  PAGE, 

(282)  (283).  Resolved  from  (2SO) 211 

Application  of  Kutter's  Formula  for  finding  velocity 211 

g For  finding  diameter  of  pipe 213 

c For  finding  coefficient  of  roughness 215 

a For  finding  sine  of  slope 216 

(159) 37-Length  due  head,  discharge  and  diameter 155 

42— Modifications  for  degrees  of  foulness  of  pipes 169 

(189) 45— Mean  pressure  in  pipe 179 

(190) 47— Mean  ordiuate  due  head  and  hydrostatic  ordinate 181 

(203) 48— Mean  pressure  per  square  inch  for  entire  pipe 181 

(204) 49— Minimum  diameter  of  an  inverted  siphon 181 

(205) 50— Minimum  thickness  of  an  inverted  siphon 182 

(2 11) 51  -Minimum  weight  of  an  inverted  siphon 182 

52— Most  suitable  form  of  a  canal 192 

(88) —Orifice  discharge  (circular) 61 

(90) —Orifice  discharge  (semi-circular—upper)  61 

(91) —Orifice  discharge  (semi-circular—lower) 62 

26— Orifice  discharge  (rectangular) 72 

(92) 27-Orifice  discharge  (rectangular) 73 

(187) 43— Pressure  per  square  inch  due  he?d 170 

(2) —  Radius  of  the  earth  at  sea  level,  due  lat 2 

£6  —Rule  for  velocity  and  discharge  of  an  open  stream  of  water 230 

(14) 5  -Time  due  velocity  and  gravity 8 

(15) 6  -Time  due  head  and  gravity 8 

(188) 44— Thickness  due  radL.s,  pressure  and  modulus  of  strength 171 

(9) 3— Velocity  due  time  and  gravity 7 

(13) 4— Velocity  due  head  and  gravity 7 

(24) 7— Velocity  through  a  rectangular  orifice 16 

(25) 8— Velocity  through  a  rectangular  orifice 16 

24) 9— Velocity  over  a  weir 18 

(25) 10— Velocity  over  a  weir 18 

(128) 33-Velocity  through  short  pipe*; 12  J 

(184) —Velocity  through  nozzles 

53— Velocity  in  a  triangular  or  rectangular  flume 199 

(28) 11-Weir  discharge  (rectangular) 19 

(30) 12-Weir  discharge  (corrected) 25 

(31) 13-Weir  discharge  (crest  three  feet  wide) 28 

(37) 14— Weir  discharge  (quadrantal) 33 

15 -Weir  discharge  (equilateral) 35 

(34)  (48) . .  16-  -Weir  discharge  (triangular) 41 

(54)  (55) . .  17— Weir  discharge  (rectangular) 44 

(61) 18  -Weir  discharge  (semi-circular) 47 

(62)  19 -Weir  discharge  (semi-circular) 48 

(63) 20— Weir  discharge  (semi-circular) 48 

(72) 21— Weir  discharge  (parabolic  open) 51 

(73) 22— Weir  discharge  (parabolic  open) 52 

(78) 24-Weir  discharge  (triangul ar  submerged) 54 

81) 25- Weir  discharge  (triangular  submerged) 58 


CONTENTS. 


PRINCIPLES  OF  HYDRA  ULICS. 

Hydraulics  Defined.— Gravity  the  source  of  motion.— Acceleration  of  gravity.— Varia- 
tion in  intensity  of  gravity  in  different  latitudes  and  at  different  altitudes.— An- 
alytical determination  of  the  laws  of  hydraulics.—  Formulas  and  rules  corre 
spending.— Examples  and  calculations.  ....  pp.  1-11 

OPENINGS. 

Submerged  and  Weir.—  Diagrams.— Analysis  of  Flow.—  Comparisons  of  formulas,  flow 
through  orifices:  rectangular  (pp.  11-16),  triangular,  trapezoidal,  circular  and 
semi-circular,  parabolic.— Examples  and  calculations.  .  .  pp.  46-73 

WEIRS. 

Discharge  over  rectangular,  triangular,  quadrantal,  equilateral,  trapezoidal,  circular, 
semi-circular,  parabolic  (pp.  18-52).— Correction  (p.  19).— Correction  due  variable 
coefficient  (p.  25).— Discharge  over  crest  three  feet  wide  (p.  28).—  Examples  and 
calcul  ations. 

MINER'S  INCH. 

Statutory.— North  Bloomfield.— Smartsville.—  South  Yuba  Canal.  -  Examples  and  cal- 
culations. .........  pp.  77-79 

CONTRACTION. 

Partial,  for  circular  and  rec' angular  orifices.— For  given  orifice  and  given  head  of 
water.— Imperfect.  .......  pp.  84- §2 

SHORT  TUBES. 

Cylindrical,   convergent,  divergent  and  compound.— Examples  and  calculations. 

pp.  102-108 

PIPES. 

Analysis  of  flow  through  pipes  (pp.  112- 116).— Empirical  formulae  coefficients  of  re- 
sistance as  found  by  Weisbach,  Darcy  and  Fanning  (pp.  116-124).—  Interpolation 
in  Table  16.  —  Flow  through  pipes -short,  long. —  Analytical  determination  of 
maximum  work,  etc.— Inlet  head.— Equations  and  rules  for  velocity,  head, 
length,  diameter  and  volume  of  flow  for  clean  iron  pipes.—  Coefficient  of  flow.— 
Effect  of  bends  —  angular,  curved.— Relative  carrying  capacities  of  clean,  foul  and 
very  foul  pipes.— Pressure  ordinatea.— Requisite  thickness  of  pipe  to  withstand  a 
given  pressure.—  Examples  and  calculations.  ...  pp.  124  - 179 


X  CONTENTS. 

NOZZLES. 

Flow  of  water  through.—  Coefficient  of  flow  due  convergence.—  Examples  and  calcula- 
tions. .  .  .  i'.'i  .  .  .  .  pp.  166-169 

INVERTED  SIPHON. 

D. termination  of  minimum  values  with  respect  to  pressure,  diameter,  thickness  and 
weight.— Hydrostatic  pressure  below  the  level  of  outlet.— Special  verification  of 
formulas.  .  ......  . ,  .  .  PP.  172  - 183 

OP^V  STREAMS. 

Flow  of  water.— Form  of  rectangle  of  maximum  carrying  capacity.— Form  of  trape- 
zoid  of  maximum  carrying  capacity.— Most  appropriate  form  of  canal. -Formulas 
for  the  flow  of  water  in  open  streams.—  Kutter's  velocity  of  water  in  a  flume  whose 
hydraulic  mean  depth  is  equal  to  that  of  a  pipe  of  given  diameter.—  Coefficient  of 
roughness  involved  in  Table  17.— Discussion  of  Kutter's  foimula.— Application  of 
Kutter's  formula  as  rendered  by  Equations  (2S2  and  283).— Application  of  Tables 
27,  28  and  29.— Interpolation  in  Table  27.—  Mean  velocity  corresponding  to  central 
surface  velocity.— Examples  and  calculations.  .  .  .  pp.  187-239 

MINING. 

Hydraulic.— Drift.— Duty  of  miner's  inch.— Examples  and  calculations,    pp.  241  -  243 

IRRIGATION. 

Quantity  of  water  required.  —  Unit  of  measure.         ....  p.  244 

Measurement  of  the  power  of  water  as  a  motor.  ....          p.  246 

Flood-flow  of  streams.  «  .  .  .  .  .  .  p.  247 


TABLES. 


No.  PAGE. 

9  ...  Coefficients  for  Flow  of  Water  Through  Circular  Orifices 85 

10 ....  Corrections  of  the  Coefficients  of  Flow  for  Circular  Orifices 94 

11 ...  .Corrections  of  the  Coefficients  of  Flow  for  Rectangular  Orifices 95 

12. ...  Corrections  of  the  Coefficients  of  Flow  Under  Head  of  Water  in  Motion . .        100 
13. . .  .Coefficients  of  Discharge  and  Velocity  for  Flow  Through  Conically  Con- 
vergent Tubes 105 

14 ....  Coefficients  of  Di  scharge  Through  Divergent  Tubes 107 

15 ....  C  oefficients  of  Discharge  Through  Compound  Tubes 109 

16. . .  .Coefficients  of  Resistance  to  the  Flow  (Cf)  of  Water  in  Clean  Pipes 113 

19. ...  Coefficients  for  Bend  Resistances  in  Pipes 161 

20. . .  .Coefficients  for  Bend  Resistances  in  Pipes  with  Circular  Transverse  Sec- 
tions   163 

21 — Coefficients  for  Bend  Resistances  in  Pipes  with  Rectangular  Transverse 

Sections . . 164 

26....Coefficients(n)  for  Roughness  of  Stream  Beds 196 

25. . .  .Dimensions  of  the  Most  Suitable  Forms  of  Canals 191 

28. . .  .Dimensions  of  Water- Ways,  etc 227-228 

2. . .  .Flow  for  Given  Depths  Over  Each  Linear  Foot  of  a  Rectangular  Weir. ...         24 

3. . .  .Flow  for  Given  Depths  with  Crest  Three  Feet  Wide 28 

4. . .  .Flow  for  Given  Depths  Over  a  Quadrant  Weir 32 

7.... Flow  of  Water  Per  Second  Through  Rectangular  Orifices    *    *    *    and 

Coefficients 69-70 

8. ...Flow  of  Water  Per  Second  Due  " Miners' Inches" 80 

22. . .  .Flow  of  Water  Through  Nozzles 167-168 

31 ....  Limiting  Velocities  in  Open  Streams 247 

23. ..  .Moduli  of  Strength,  Working  Load  and  Safety 172 

30 ....  Mean  Velocity  from  Central  Surface 240 

32 ....  Miscellanies 249-252 

24. ..  .Number,  Thickness  and  Weight  of  One  Square  Foot  of  Sheet  Iron 184 

27.... Open  Streams— Flow  of  for  Coefficients  of  Roughness  n=.012,  n=.017, 

n=.025  and  n=.035 219  -226 

29.... Relations  of  Depth,  Base  and  Slope  of  Bank 229 

5. . .  .Square  Roots,  Cubes  of  Square  Roots  and  Fifth  Powers  of  Square  Roots 

of  Numbers 65 

6. . .  .Square  Roots  of  Numbers 66-67 

17.... Velocities  and  Quantities  of  Flow  in  Clean  Iron  Pipes  Due  Given  Slopes 

and  Diameters 134-143 

18 ....  Velocities  in  Pipes  and  Corresponding  Inlet  Heads . . . .' 151 

1. . .  .Weir  Coefficients ." 22 


WNWERS1TY 

OF 


PRINCIPLES  OF  HYDRAULICS. 


The  term  hydraulics  was  originally  applied  to  water 
pipes,  or  to  the  conveyance  of  water  through  pipes. 
It  is  now  used  in  a  wider  sense  to  designate  that 
branch  of  engineering  which  treats  of  water  in  mo- 
tion, the  means  of  measuring,  conveying,  and  raising 
it,  and  its  application  to  machinery  as  a  prime  motor. 

The  source  of  this  motion  is  the  force  of  gravity — 
a  force  which  acts  indiscriminately  upon  every  parti- 
cle of  matter,  and  impresses  upon  each  particle  at 
every  instant  the  same  degree  of  velocity  in  vacuo. 
The  fundamental  principles  or  laws  of  hydraulics  then, 
are  those  of  uniformly  varied  motion. 

'These,  with  respect  to  a  body  falling  from  rest 
through  a  certain  hight  in  vacuo,  are  as  follows : 

1st — The  velocities  acquired  are  proportional  to  the 
times  elapsed  since  the  beginning  of  the  motion. 

2d — The  spaces  fallen  are  proportional  to  the 
squares  of  the  times  elapsed. 

3d — These  spaces  or  hights  are  proportional  to  the. 
squares  of  the  velocities  acquired  at  the  end  of  each. 

4th — The  velocity  acquired  at  the  end  of  the  first 


2  PRACTICAL   HYDRAULICS. 

unit  of  time  is  equal  to  twice  the  distance  fallen  dur- 
ing this  time. 

The  intensity  of  the  force  of  gravity  varies  in  dif- 
ferent latitudes,  but  for  most  purposes  in  hydraulic 
calculations,  it  may  be  regarded  constant. 

The  velocity  which  the  force  of  gravity  can  gener- 
ate in  a  second  of  time  at  the  surface  of  the  earth,  is 
usually  designated  by  g,  and  is  termed  acceleration  of 
gravity.  Its  value,  as  given  in  Rankine's  Applied  Me- 
chanics, p.  485,  for  Lat.  45°  at  sea-level,  is  g. 

Thus  #=32.1695  feet.  (1) 

For  the  most  part  in  practice  #  =  32.2. 

In  case  a  high  degree  of  accuracy  is  required,  the  ob- 
lateness  of  the  earth,  the  latitude  and  elevation  of  the 
place  at  which  the  value  of  the  force  of  gravity  is 
sought,  have  to  be  considered. 

The  formulas  involving  these  elements  (the  two  for- 
mer deduced  from  numerous  pendulum  experiments 
made  in  various  parts  of  the  world),  are  given  in 
Rankine's  Applied  Mechanics,  pp.  485-486,  as  follows: 

£=20,887,540  feet  (l  +  0.00164  cos  2A  (2) 

g=g(l— 0.00284  cos  2&)  (l— 2£)  (3) 

In  which  R  denotes  the  earth's  radius  at  the  lo- 
cality of  observation,  I  the  latitude,  g'  the  force  of  grav- 
ity at  sea-level,  in  Lat.  45°,  and  g  the  force  of  gravity  for 
the  elevation  above  sea-level  in  the  given  Lat.  1.  The 


PRACTICAL   HYDRAULICS.  o 

factor  M }  in  the  second  member  of  equation  (3)  is 

readily  obtained  from  the  well-established  proposition 
that  the  gravity  of  a  body  varies  inversely  as  the 
square  of  the  distance  from  the  center  of  the  earth. 
If  in  a  given  latitude  I,  g  denote  the  force  of  gravity 
at  sea-level,  and  g"  the  force  of  gravity  at  an  eleva- 
tion, h,  there  will  result: 

g»=g    (l_*)  (4) 

y         V      \          R) 

When  h=Q,  it  is  obvious  that  in  equations  (3)  and 
(4),  the  factor  (l—  ^)=1. 

When  Z=45°,  cos  21=0. 

Whence,  (2),  #^20,887,540  feet—3956  miles.    (2  ) 

Example  1. — What  is  the  value  of  the  force  of 
gravity  at  Presidio,  San  Francisco,  in  latitude  37°, 
47',  48",  at  sea-level? 

Calculation. — Employing  formula  (3). 

Find  cos  2  (37°,  47',  48",)=0.2488. 

Substitute  this  value,  namely,  0.2488.  and  the  value 
of  #'=32.1695,  of  equation  (1)  in  equation  (3),  ob- 
serving that  &=0,  #=32.1695  (1—0.00284X0.2488). 
Whence,  #=32.1468  ft.— Answer. 

Ex.   2.— In  latitude    37°,    47',    48",  as    in   Ex.   1, 
what  is  the  force  of  gravity  at  an  elevation  of  2  miles? 
Cal.—WQ  find  from  (2)  and  (2,). 
#=3956   (1+0.00164X.2488). 
Whence,  #=3957.6  miles. 


4  PRACTICAL    HYDRAULICS. 

Substituting  this  value  of  R  and  the  value  of  g  as 

x  2  x  2  \ 

found  from  Ex.  1,  inequation  4,  #"  =  32.1468  (l— 3^r0) 
Reducing  #"^32.1144.— Ans. 

Ex.  3. — Required  the  force  of  gravity  at  an  eleva- 
tion of  two  miles  above  sea-level  at  the  equator. 

Col.— When  1=0,  find  cos  2J=1.  Substituting 
this  value  of  21  in  eqs.  (2)  and  (2J,  and  reducing,  .R--= 
3962.5.  Substituting  the  value  of  R=--  3962.5,  the 
value  of  cos  2Z=1,  the  value  of  #'=-32. 1695,  and 
the  given  value  of  h  —  2  miles  in  formula  (3), 

#=32.1695  ( 1—0.00284")  (l  —  —} 

\  )     \  S962.5/ 

Whence,  #--32.0457.— Ans. 

Remark. — Had  we  made  R=  4000  in  the  preceding 
examples,  it  would  in  no  case  have  varied  the  result  to 
exceed  .0003.  We  may,  therefore,  without  sensible 
error,  regard  the  radius  constant  and  equal  to  R=-  4000. 

These  refinements,  with  respect  to  variations  in  the 
force  of  gravity  under  different  conditions,  though 
highly  important  in  establishing  a  standard  of 
measurement,  and  in  various  scientific  investigations, 
yet  for  the  most  part  are  little  applied  in  practice. 

Reverting  to  the  subject  of  the  laws  of  varied  mo- 
tion, it  will  be  noted  that  the  velocity  acquired  by 
a  falling  body  at  the  end  of  the  first  second  of  time,  is 
double  the  hight  which  the  body  has  fallen  during 
that  time.  A  body  then,  in  vacuo,  falls  during  the 
first  second  of  time  16.1  feet,  or  more  accurately,  16.085 
in  Lat.  45°. 


PRACTICAL    HYDRAULICS.  O 

Denote  in  seconds,  the  times  of  a  falling  body  in 
vucuo  by  the  consecutive  numbers: 

1,  2,  3,  4,  5,   6,  7,  8,  9,  etc. 

Then,  according  to  the  2d  law,  the  hights  of  the 
fall  are  proportional  to  the  squares  of  these  times; 
thus— 1,  4,  9,  16,  25,  36,  49,  64,  81,  etc.;  and,  accord- 
ing to  the  3d  law,  the  hights  are  proportional  to  the 
squares  of  the  velocities  acquired  during  these  times. 

If  the  hight  of  fall,  as  found  by  law  2d,  due  any 
given  time,  be  taken  from  the  hight  of  fall  due  this 
time  increased  by  one  second,  the  remainder  will  be 
equal'  to  the  space  fallen  during  this  increment  of, 
one  second. 

Thus  the  hights,  so  fallen  in  the  natural  order  of 
times,  are  1,  3,  5,  7,  9,  11,  13,  15,  17,  etc. 


EXAMPLES  AND  CALCULATIONS. 

Ex.  4. — How  many  feet  will  a  body,  as  water  in 
vacua,  fall  during  the  5th  second  of  its  descent? 

Gal. — By  the  foregoing  it  will  be  seen  that  the  fall 
during  the  1st  second  is  16.1  feet  nearly,  and  during  the 
5th  second  is  9  times  as  much ; 

Hence,  16.1  X  9-144.9  feet.— Ans. 

Ex.  5. — What  distance  will  water  fall  in  vacuo  in 
5  seconds? 

Gal. — Note  in  accordance  with  law  2d,  that  a  body 


6  PRACTICAL   HYDRAULICS . 

falls  25  times  as  far  in  5  seconds  as  it  does  in  1  second ; 
hence,  16.1X25=402.5  feet.— Ans. 

In  further  illustration: 

Gal. — Note  that  the  hights  fallen  respectively  dur- 
ing each  second  of  the  given  time  of  5  seconds,  are 
1,  3,  5,  7,  9.  The  sum  of  which  is  25,  as  found  in  the 
preceding  calculation;  hence,  16.1X25=402.5  feet. 

Ex.  6. — "What  will  be  the  velocity  of  water  falling, 
in  vacuo,  at  the  end  of  the  5th  second? 

Gal. — Observe  that,  according  to  the  1st  law,  the 
velocity  is  5  times  as  great  at  the  end  of  the  5th  sec- 
ond as  it  was  at  the  end  of  the  first — that  is,  2X  5=10; 
hence,  16.1X10=161.  feet. 


RULES  WITH  RESPECT  TO  THE  RELATIONS  OF  SPACE, 
TIME,  VELOCITY,  AND  THE  FORCE  OF  GRAVITY,  INVOLVED 
IN  THE  FALL  OF  A  BODY,  AS  WATER,  IN  VACUO. 

The  velocity  given  to  find  the  head  or  distance  the 
ivater  has  fallen. 

Rule  1 . — Divide  the  square  of  the  given  velocity  by 
twice  the  acceleration  of  gravity — that  is,  by  64.4. 

Ex.  7. — The  velocity  is  150  feet  per  second.  What 
is  the  head  or  distance  fallen? 

Cal— 150X  150-^-64.4=349.4  feet  nearly.— Ans. 

To  find  the  head  or  distance  water  will  fall  in  a 
given  time 


PEACTICAL  HYDRAULICS,  t 

Rule  2. — Multiply  the  square  of  the  time  in  sec- 
onds by  16.1  feet. 

Ex.  8. — What  distance  will  water  fall  in  4  seconds? 
tfaZ.— 4X4X  16.1=  257.6  feet.— Ans. 

The  time  given  to  find  the  velocity  of  water  fatting 
freely. 

Rule  3. — Multiply  the  time  in  seconds  by  the  accel- 
eration of  gravity,  namely,  32.2  feet. 

Ex.  9. — What  velocity  does  water  acquire  in  falling 
7  seconds? 

(7aZ.— 32.2X7=225.4  feet.— Ans. 

The  head,  or  distance  of  water  fallen  freely,  given 
to  find  the  acquired  velocity. 

Rule  4. — Extract  the  square  root  of  twice  the  pro- 
duct of  the  head  and  the  acceleration  of  gravity,  or 
multiply  the  square  root  of  the  given  head  by  the 
square  root  of  twice  the  acceleration  of  gravity — 
that  is,  by  8.025. 

Ex.  10. — What  velocity  will  water  acquire  by  fall- 
ing freely  196  feet?  In  other  words,  with  what  ve- 
locity will  it  flow  under  a  pressure  or  head  of  196 
feet? 

Cal. — Square  root  of  196  feet  is  14  feet: 

Then  14x8.025-112.35  feet.— Ans. 


PRACTICAL    HYDRAULICS. 

The  velocity  being  given  to  find  the  time  which  a 
body,  as  water,  has  fallen. 

Rule  5. — Divide  the  given  velocity  by  the  acceler- 
ation of  gravity,  viz.,  32.2  feet. 

Ex.  11.— The  velocity  is  322  feet  per  second.  What 
is  the  time  fallen? 

Cal— 322-^32.2=10  seconds.— Ans. 

To  find  the  time  required  for  water  to  fall  freelij 
through  a  given  distance. 

Rule  6. — Extract  the  square  root  of  twice  the  given 
distance,  divided  by  32.2,  or,  in  other  words,  divide 
the  square  root  of  the  given  distance  by  the  square 
root  of  one-half  the  acceleration  of  gravity — that  is, 
by  4.012. 

Ex.  12. — What  time  will  water  require  to  fall  freely 
a  distance  of  144  feet? 

Cal. — The  square  root  of  144  feet  is  12  feet: 
Hence,  12-^-4.012=3  seconds  nearly. — Ans. 

Determination  of  these  laws  of  hydraulics  by 
analysis. 

To  determine  these  laws  let  h  represent  the  head  or 
distance  in  feet,  through  which  the  water  acts,  or  has 
fallen  in  a  given  time  denoted  by  £;  let  v  represent  the 
velocity  acquired  at  the  bottom  of  this  head  or  dis- 
tance fallen,  and  let  g  denote  the  acceleration  of 


PRACTICAL    HYDRAULICS. 

gravity — that  is,  the  velocity  which  the  force  of 
gravity  can  generate  in  a  second  of  time  at  the  sur- 
face of  the  earth. 

Then  the  expressions  for  velocity  and  acceleration 
of  gravity  will  be: 

(5) 

dt  ^    } 

and  0=!  (6) 


Eliminate  dt  from  (5)  and  (6); 


-  (7) 

9 


Integrating  (7),  fc=£  (8) 

Integrating  (6),  v^cjt  (9) 

Combining  (5),  and  (9),  dh^gtdt  (10) 

Integrating  (10),  h  =  ~  (11) 

Combining  (9),  and  (11),  fc=~  (12) 

Reduce  (7),  as  to  v, 

v=i/w-=i/lXi/h  (13) 

Divide  (9)  by  g,  t=-  (14) 

^  • 

Reduce  (11),  as  to  tt  t=\/  '  h~^V\  (^) 

By  inspection  it  will  be  seen  that  in  the  text,  the 
given  rules  and  the  enunciations  termed  laws  are  de- 
rived from  these  formulas,  or  are  but  expressions  for 


10  PRACTICAL  HYDRAULICS. 

them ;  and  that  the  relations  of  space,  time,  velocity 
and  force  of  gravity  are  more  clearly  expressed  by 
formula  than  it  was  possible  to  do  by  words. 

To  facilitate  this  inspection,  the  following  references 
to  the  different  forms  of  expression,  but  equivalent  in 
meaning,  are  given.  Thus: 

Laws.         Rules.  Formulas. 

1st.          3d.          (9),  v=gt. 

2d.  2d.          (11),  .  .  '          ft=£ 

3d.          1st.          (8),  h=9- 

4th.  Modification  of  (12),    v=2fc-5-l  Sec. 

4th.  (13),  reduced  from  (8),          v=i/zjh 

5th.  (14),  reduced  from  (9),  t-= 

6th.  (15),  reduced  from  (11),          *= J- 

\  g 

The  value  of  g  being  substituted  in  these  formulas, 
every  possible  question  with  respect  to  the  free  fall  of 
water  or  other  body  can  be  answered. 

The  formula  of  most  frequent  occurrence  in  hy- 
draulics corresponds  to  the  3d  law,  1st  rule,  and  is 
denoted  in  the  column  of  formulas  above  by  (8) — that 

is,  h—-.  This  formula,  reduced  with  respect  to  v,  is 
denoted  by  (13);  v=i/ii=i/£Xi/A=8.025i/ji,  in 
which  forms  it  frequently  occurs  in  works  in  en- 


PRACTICAL  HYDRAULICS.  11 

gineering.  In  this,  or  these  formulas,  v  is  termed  the 
velocity  due  to  a  given  hight,  h,  and  h  the  hight  due 
to  a  given  velocity,  v — that  is,  v  denotes  the  distance 
which  water  flowing  freely  under  a  pressure  of  h  feet 
in  hight,  will  pass  over  in  one  second  of  time  at  the 
bottom  of  this  hight,  h,  which  velocity  is  the  same, 
as  water  falling  freely  through  the  hight,  h,  would 
acquire  at  its  bottom. 

FLOW  OF  WATER  THROUGH  OPENINGS. 

Openings  are  of  two  classes — the  submerged  and 
the  weir. 

An  opening  having  its  top,  as  shown  at  B,  Fig.  1, 
beneath  the  water's  surface,  AD,  is  termed  a  sub- 
merged orifice ;  an  opening  having  an  open  top,  as 
shown  at  C,  the  crest,  in  Fig.  2,  is  termed  a  weir. 
In  both,  the  form  of  outlet,  for  the  most  part  in  prac- 
tice, is  rectangular.  This  is  more  especially  true  with 
respect  to  weirs.. 

In  Fig.  1,  representing  a  vertical  section  through  a 


FIG.  I. 

FIG.  2. 

rectangular  opening,  conceive  the  opening,  BC,  com- 
posed of  horizontal  fluid  layers  indefinitely  thin. 


12  PRACTICAL  HYDRAULICS. 

Let  £-—  horizontal  length  of  opening. 
&  =AB  head  with  respect  to  top  of  opening. 
h  =  AC  head  with  respect  to  bottom  of  opening. 
#=head  additional  to  h,   and  due  any  horizontal 
fluid  layer  indefinitely  thin  in  the  opening  BC. 
dx=  thickness  of  such  fluid  layer. 
v=  velocity  due  (h-\-x)  per  second. 
Q=  discharge  of  water  in  cubic  feet  per  second. 

Then  by  (13),  v=(2g)  $  (h+x)  *  (16) 

dQ=l  (20)  i  (h+x)  i  dx.  (17) 

Integrating  (17),  between  the  limits  of  x==Q,  and 
x=h—  h--=EG. 

(18) 


Equation  (18),  expresses  the  quantity  of  water,  in 
cubic  feet,  which  will  flow  through  a  submerged  ori- 
fice under  the  general  conditions  imposed.  It  is, 
however,  alike  applicable  to  weirs. 

For,  by  making  &=0,  there  results: 


(19) 

which  expresses  the  quantity  of  water  in  cubic  feet 
that  will  flow  over  a  weir  under  the  general  condi- 
tions imposed. 

The  true  mean  velocity  of  a  stream  through  the 
submerged  orifice  under  the  given  conditions,  is, 


(20) 


PRACTICAL   HYDRAULICS.  1.3 

Some  hydra  ulicians  assume  that  the  mean  head  is 
equal  to  the  distance  between  the  surface  and  the  mid- 
dle of  the  submerged  orifice;  hence,  deduce  that  the 
mean  velocity  is  by  (13), 


Comparing    (20),  and   (21),    omitting   the  common 
factor  (2#)i 


(22) 


In  refutation  of  the  assumption  that  the  mean  ve- 
locity of  a  stream  of  water  flowing  from  a  sub- 
merged orifice,  takes  place  at  the  middle  of  the  open- 
ing, the  following  illustrations  are  presented.  By 
substituting  the  respective  values  of  the  given  heads 
and  vertical  widths  in  equation  (22),  there  result: 

When  the  vertical  width  is  equal  to  one-half  the 
head  above  it,  the  velocity  is  found  one-  sixth  of  one 
per  cent  too  great  ;  when  it  is  equal  to  the  head  above 
it,  the  velocity  is  found  one-half  of  one  per  cent  too 
great; 

When  it  is  equal  to  twice  the  head  above  it,  the 
velocity  is  found  one  and  one-ninth  per  cent  (.0111) 
too  great; 

When  it  is  equal  to  three  times  the  head  above  it, 
the  velocity  is  found  one  and  two-thirds  of  one  per 
c«nt  (.01645)  too  great; 

And   when  we  pass  to  the  limit  of   the  weir,   by 


14  PEACTICAL  HYDRAULICS. 

making  h=0,  the  velocity  becomes  six  per  cent  too 
great. 

The  assumption  that  the  mean  velocity  takes  place 
at  the  middle  of  a  rectangular  opening,  is  thus  shown 
erroneous.  It  may,  perhaps,  approximate  the  truth 
with  sufficient  accuracy  for  most  practical  purposes, 
when  the  vertical  width  of  the  opening  is  less  than  the 
head  above  it;  but  is  inadmissible  when  the  width  in 
comparison  is  greater. 

The  formula  obtained  on  the  assumption  that  the 
mean  velocity  takes  place  at  the  middle  of  the  open- 
ing, is  commended  by  its  simplicity  of  expression. 
But  its  application,  in  case  the  ratio  of  the  head  to  the 
vertical  opening  is  large,  involves  the  use  of  a  co- 
efficient of  flow,  varying  with  that  ratio.  The  for- 
mula so  encumbered  would  evidently  be  more  complex 
and  tedious  of  application  than  that,  namely  (20), 
which  it  seeks  to  evade  or  displace. 

Water  in  motion  is  subject  to  resistances,  which 
retard  its  velocity  and  diminish  its  volume  of  flow. 
Modifications,  therefore,  are  requisite  to  be  made  in 
the  theoretical  formulas  of  hydraulics,  so  that  they 
shall  embrace  'the  measure  of  these  resistances,  and 
thereby  more  fully  meet  the  requirements  of  practice. 
These  modifications  are  effected  for  the  most  part  by 
coefficients,  whose  values  have  been  determined  by 
experiment.  A  coefficient,  as  here  used,  corresponds 
to  that  part  of  the  theoretical  quantity,  as  velocity 
and  volume,  remaining  after  it  has  been  diminished  in 
amount  equal  to  the  loss  due  resistances  overcome. 


PRACTICAL  HYDRAULICS.  15 

With  respect  to  the  velocity  of  water  flowing  through 
orifices,  a  multiplicity  of  experiments,  during  a  period 
of  many  years,  have  been  made  with  extreme  care  by 
the  ablest  hydraulicians.  The  tabulated  results  of  these 
experiments,  differing  with  respect  to  head  and  size  of 
opening,  vary  from  .572  to  .795. 

If  the  theoretical  velocity  of  water  flowing  through 
a  rectangular  opening  with  thin  sides  or  lips  be  taken 
as  a  unit,  then  will  an  average  of  the  experiments 
referred  to,  closely  approximate  .62. 

This  fraction  is  very  generally  adopted — that  is,  for 
the  entire  opening,  the  theoretical  velocity  is,  to  the 
average  experimental  velocity,  as  100  to  62. 

Let  it  not  be  inferred  that  the  actual  velocity  of  the 
flowing  water  is  62  per  cent  of  the  theoretical  ve- 
locity. Water,  in  flowing  from  a  reservoir,  ap- 
proaches the  opening  or  outlet  in  convergent  lines. 
This  convergence  continues  a  short  distance  beyond 
and  outside  the  outlet,  as  shown  at  M  M/?  Fig.  1, 
where  occurs  the  minimum  cross  section  of  the  stream, 
and  where  the  velocity  is  nearly  equal  to  the  theoreti- 
cal velocity.  The  area  of  the  outlet  being  taken  as 
the  unit,  the  area  of  this  cross  section  is  equal  to 
.637,  while  the  velocity  of  the  stream  at  this  point  is 
equal  to  .974  of  the  theoretical  velocity.  The  co- 
efficient of  discharge  there  is  equal  to  the  product  of 
these  coefficients  of  cross  section  and  velocity.  Let 
c=this  coefficient  of  discharge; 

Then  c= .637  X .974^.62  nearly.  (23) 


16  PRACTICAL   HYDRAULICS. 

In  weir  openings,  the  experiments  of  J.  B.  Francis, 
C.  E.,  make  c=.622.  The  top  contraction  seems  to 
have  no  separate  coefficient  in  the  formula  volume. 
C,  the  coefficient  of  discharge,  is  also  the  coefficient 
of  average  velocity,  at  and  for  the  entire  opening,  but 
not  as  hitherto  remarked  for  obtaining  the  actual  ve- 
locity, as  it  occurs  at  M  My,  Fig.  1.  Introducing  this 

coefficient  (.622),  or  modifier,  (2^)^  into  equations  (20) 

and  (21),   and  making  (%i)^=8.026  as  found,  there 
results, 


(25) 


TO    DETERMINE   THE   VELOCITY    OF   WATER    FLOWING 
THROUGH  A  RECTANGULAR    OPENING. 


Rule  7. — From  the  square  root  of  the  cube  of  the 
head  of  water  on  the  bottom  of  the  opening,  subtract 
the  square  root  of  the  cube  of  the  head  on  the  top  of 
the  opening.  Divide  the  remainder  by  the  difference 
of  these  heads,  and  multiply  this  quotient  by  3.33, 
the  product  will  be  the  required  velocity.^ 

Rule  7. — Corresponds  to  equation  (24). 

Rule  8. — Multiply  the  square  root  of  one-half  the 


PRACTICAL    HYDRAULICS. 

sum  of  the  respective  heads  on  the  top  and  bottom  of 
the  opening,  by  4.99. 

Rule  8. — Corresponds  to  equation  (25). 

Ex.  13. — In  a  rectangular  opening,  the  head  on  the 
top  of  the  orifice  is  2.25  feet,  and  on  the  bottom  of  the 
opening  is  4  feet.  What  is  the  average  velocity  of  the 
flow  of  water  at  the  outlet? 

Gal.  by  Rule  7,  or  formula  (24). 

Square  root  of  the  cube  of  the  head  on  the  bottom  of 
the  opening  (4)i~=8; 

Square  root  of  the  cube  of  the  head  on  the  top  of 
the  opening  (2.25)1=3.375. 

Difference  between  heads  4 — 2.25=1.75. 

Thus:     3.33  (8.— 3.375)-^-1.75=8.8  feet.— Ans. 

Gal.  by  Rule  8,  or  formula  (25). 

Half  sum  of  given  heads  (4+2.25)^-2=3.125; 
4.99  times  the  square  root  of  this  quotient. 

4.99  (3.125)^=8.814  feet.— Ans. 

Dividing  the  result  obtained  under  Rule  8  by  result 
obtained  under  Rule  7. 

8.814-5-8.8=1.0016, 

It  is  seen  that  in  this  case,  Rule  8,  corresponding  to 
formula  (25)  gives  a  velocity  nearly  one-sixth  of  one 
per  cent  too  great. 


18  PRACTICAL    HYDRAULICS. 

TO  FIND  THE  AVERAGE  VELOCITY  OF  WATER  FLOWING 
OVER  A  WEIR.    • 


Rule  9. — Multiply  the  square  root  of  the  head  over 
the  crest  by  3.33. 

Rule  9  corresponds  to  formula  (24)  by  making  the 
head  on  the  top  nothing. 

Rule  10. — Multiply  the  square  root  of  one-half  the 
head  over  the  crest  by  4.99. 

Rule  10  corresponds  to  formula  (25)  by  making 
the  head  on  the  top  nothing. 

Ex.  14. — In  a  rectangular  outlet,  open  top,  the 
head  on  the  bottom  of  the  opening  is  one  foot.  What 
is  the  average  velocity  of  the  flow? 

Gal.—  By  Rule  9. 

Square  root  of  the  given  head  (1)^=1. 

Then  3.33X  1=3.33  feet.— ^s. 

Gal.—  By  Rule  10. 

Square  root  of  one-half  the  given  head: 

(.5)i=.7072. 

Then  4.99X  .7072=3.53  feet.— Ans. 
Dividing  result  obtained  under  Rule  10  by  that  ob- 
tained under  Rule  9, 

3.53-^3.33=1.06. 

It  is  hereby  seen  that  Rule  10  gives  a  velocity  six 
per  cent  too  great. 


PEACTICAL    HYDRAULICS.  19 

Substituting  the  values,  of  (2^)2=8.025, 
and  c=. 622  of  (23)  in  (18). 

Q=3.33Z  (hp—h*  (26) 

When  h—Q,  (26)  becomes 

Q=3.33Z  h*  (2T) 

Formula  (26)  is  adapted  to  finding  the  discharge  of 
a  rectangular  submerged  orifice,  and  formula  (27),  the 
discharge  of  a  weir.  In  the  latter  case,  when  the 
depth  of  water  on  the  crest  exceeds  three  inches,  and 
does  not  exceed  two  feet,  and  the  length  of  weir  is  not 
less  than  three  times  this  depth,  J.  B.  Francis,  C.  E., 
a  most  eminent  experimentalist,  determined  by  care- 
ful experiments  made  on  a  large  scale,  and  under  the 
most  favorable  circumstances,  that  the  loss  by  means 
of  end  contraction,  is  equal  to  one-tenth  the  depth  of 
water  over  the  weir  for  each  such  contraction. 

Introducing  this  modification  in  (27)  and  there  re- 
sults: 

Q=3.33  (£—0.1  nh)hl  (28) 

In  which  n  denotes  the  number  of  end  contractions- 
For  a  weir  one  foot  in  length  with  water  one  foot 

deep,     Nystrom's    Mechanics    makes    the    coefficient 

3. 135  instead  of  3.33. 

TO  FIND  THE  DISCHARGE  OF  WATER  OVER  A  WEIR, 
WITH  CORRECTION  MADE  FOR  DEPTH  ON  CREST. 

Rule  11. — Deduct  from  the  length  of  the  weir,  one- 


20  PRACTICAL   HYDRAULICS. 

tenth  the  depth  of  water  over  the  crest,  for  each  and 
every  end  contraction  -(usually  two) ;  multiply  the  cor- 
rected length  so  found  by  3. 33  times  the  square  root 
of  the  cube  of  the  depth  or  head  of  water  on  the 
crest. 

Ex.  15. — The  length  of  weir  being  3.01  feet,  cut 
in  two-inch  planks,  and  the  full  depth  h,  over  the  bot- 
tom of  the  notch  1.023,  what  is  the  discharge  in  cubic 
feet  per  second? 

Gal.—  By  Rule  11. 

Loss  in  this  case  by  two  end  contractions: 

1.023X.2=.2046. 

Corrected  length:     3. 01—. 2046=2. 8054  feet. 
Square  root  of  the  cube  of  the  given  head: 

(1.023)1=1.085;  hence, 

<2=3.33x2.8054Xl.035=9.67    cubic  feet.—  Ans. 

Working  this  example  by  the  weir  formula  given 
in  Weisbach's  Mechanics,  whence  the  example  is  taken, 
there  results: 

Q=10.22  cubic  feet  flow  per  second,  which  is  five 
and  two-thirds  per  cent  more  than  obtained  by  Rule 
11,  derived  from  the  formula  in  accordance  with  ex- 
periments of  Francis. 

•This  discrepancy  would  seem  to  arise  mostly  from 
the  correction  introduced  in  our  rule  to  compensate 
for  end  contraction. 


PRACTICAL    HYDRAULICS.  21 

The  length  of  weir  being  3.01  feet,  and  the  depth 
of  water  on  crest  .545  feet,  the  discharge  by  Rule  11 
or  formula  (28)  amounts  to  3.887  cubic  feet  per  sec- 
ond, and  by  Weisbach's  formula  4.253,  which  latter  is 
nine  and  one-half  per  cent  greater  than  obtained  by 
Rule  11,  or  formula  28. 

Again,  the  length  of  weir  being  3.01  feet,  and  the 
depth  of  water  on  the  crest  .189  feet,  the  discharge 
per  second,  by  Rule  11,  amounts  to  .8133  cubic  feet, 
and  by  Weisbach's  formula .  753  cubic  feet,  which  latter 
is  nearly  seven  and  one-half  per  cent  less  than  the 
discharge  obtained  by  Rule  11  or  formula  (28). 

In  the  foregoing  examples,  the  length  of  weir,  the 
respective  depth  of  water  on  the  crest,  and  the  cor- 
responding coefficient  of  discharge,  are  given  as  the 
data  and  results  of  actual  experiments  made  by  W.  R. 
Johnson,  editor  of  the  first  American  edition  of  Weis- 
bach's Mechanics. 

The  experiments  of  Professor  Johnson  are  entitled 
to  great  consideration.  Those  of  Mr.  Francis,  how- 
ever, from  which  formula  (28),  or  Rule  (11)  is  derived, 
were  conducted  on  a  large  scale  with  extreme  care  and 
with  the  aid  of  the  most  improved  mechanical  appli- 
ances, so  as  to  commend  their  results  as  the  best 
authority  known  at  present  in 'weir  measurement. 

In  addition  to  the  variation  expressed  by  the  factor 
(/ — O.Inh)  in  formula  (28),  the  coefficient  of  dis- 
charge is  found  to  vary  with  the  ,  head  or  depth  of 
water  'on  the  crest  of  the  weir. 


22 


PRACTICAL    HYDRAULICS. 


To  compensate  for  this  variation,  Table  1  is  given, 
to  be  used  with  formula  (28). 


TABLE  I. 


WEIR   COEFFICIENTS. 
Water  Supply- Engineering  J.  T.  Fanning. 


Depth  in}}°;t 

Coefficient 

12 
1.000 

3.339 

1 
.083 

3.263 

14 
1.167 

3.339 

.124 
3.274 

16 
1.333 

3.340 

2 
.167 

3.285 

18 
1.500 

3.339 

3 
.25 

3.301 

20 
1.667 

3.339 

4 
.333 

3314 

24 
2.000 

3.338 

6 
.500 

3.329 

30 
2.500 

3.334 

8 
.667 

3.336 

40 
3.333 

3.331 

10 
.833 

3.338 

48 
4.000 

3.317 

1  in 

Depth   m  }-f  '. 
j  ieezj. 

Coefficient  

The  mean  coefficient,  as  given  in  formula  (28),  is 
3.33.  The  maximum,  as  seen  by  Table  1,  is  3.34. 

The  mean  coefficient  3. 33  corresponds  to  the  depths 
523  feet  and  8.42  feet. 

A  comparison  shows  that  the  mean  coefficient  is 
three-tenths  of  one  per  cent  (.003)  less  than  the  maxi- 
mum, two  per  cent  greater  than  that  corresponding  to 
a  depth  of  one  inch  or  0.083  feet;  four- tenths  of  one 
per  cent  (.004)  greater  than  that  due  a  depth  of  four 
feet;  and  nine-tenths  of  one  per  cent  greater  than 
due  a  depth  of  one-fourth  of  a  foot.  The  greatest 
variation  occurring  in  the  coefficient  of  discharge  for 
different  depths,  between  four  feet  and  one-fourth  of  a 


PKACTICAL    HYDRAULICS. 

foot,  is  seen  to  be  below  one  per  cent.  An  equal  vari- 
ation, for  the  most  part,  in  practice,  is  likely  to  occur 
from  various  other  causes,  and  too  often  elude  the  ob- 
servation of  the  engineer. 

Ex.  16. — The  length  of  a  weir  being  six  feet,  and 
the  full  depth  of  water  over  the  crest  two  inches, 
what  is  the  discharge  per  second? 

Col.— By  formula  (28)  modified  by  Table  1. 

Loss  in  this  case  by  two  end  contractions. 

2inches=.167feet;  .  167 X.2=. 0334. 

6— .0334— 5.9666  corrected  length. 

3 

(.167)^=.  06824,  square  root  of  cube  of  given  head. 
3.285— coefficient  as  per  Table  1,  due  head  of  two 
inches. 

<2=3.285X5.9666X. 06824=1.307  cu.  feet.— Ans. 

Ex.  17.— The  length  of  weir  being  10.8  feet,  and 
full  depth  of  water  over  crest  four  feet,  what  is  the 
discharge  per  second? 

Cdl.—By  formula  (28)  modified  by  Table  1. 
10.8 — 4  X- 2=10  corrected  length  of  weir. 
(4)^=8,  square  root  of  cube  of  given  head. 

3.317=coefficient  as  per  Table  1,  due  head  of  four 
feet. 

Q=3. 317X10X8=265. 36  cubic  feet.— Ans. 


24  PRACTICAL    HYDRAULICS.       , 

TABLE  II. 

Flow  for  given   depths   over  each  linear  foot  of  a 
rectangular  weir. 


Head. 
Feet. 

Flow 
Cubic  Feet. 

Head. 
Feet. 

Flow 
Cubic  Feet. 

Head. 
Feet. 

Flow 
Cubic  Feet. 

.04 

.0261 

.46 

1.0386 

.2 

4.3904 

.05 

.0365 

.48 

1.1072 

.3 

4.9506 

.06 

.0480 

.50 

1.1771 

.4 

5.5311 

.07 

.0604 

.52 

1.2483 

.5 

6.1341 

.08 

.0737 

.54 

1.3209 

.6 

6.7558 

.09 

.0881 

.56 

.3951 

.7 

7.3987 

.10 

.1035 

.58 

.4724 

.8 

8.0516 

.11 

.1195 

.60 

.5506 

.9 

8.7317 

.12 

.1369 

.62 

.6286 

2.0 

9.4399 

.13 

.1536 

.64 

.7080 

2.1 

10.1460 

.14 

.1718 

.66 

.7888 

2.2 

10.8694 

.15 

.1906 

.68 

.8705 

2.3 

11.6189 

.16 

.2102 

.70 

.9599 

2.4 

12.3850 

.17 

.2303 

.72 

2.0380 

2.5 

13.1668 

.18 

.2512 

.74 

2.1237 

2.6 

13.9649 

.19 

.2726 

.76 

2.2118 

2.7 

14.7783 

.20 

.2951 

.78 

2.2996 

2.8 

15.6067 

.22 

.3407 

.80 

2.3883 

2.9 

16.4501 

.24 

.3882 

.82 

2.4788 

3.0 

17.3086 

.26 

.4377 

.84 

2.5699 

3.1 

18.1809 

.28 

.4892 

.86 

2.6620 

3.2 

19.0676 

.30 

.5445 

.88 

2.7557 

3.3 

19.9687 

.32 

.593:' 

.90 

5.8500 

3.4 

20.7953 

.84 

.6572 

.92 

2.9463 

3.5 

21.7194 

.36 

.7158 

.94 

3.0432 

3.6 

22.6568 

.38 

.7761 

.96 

3.1409 

3.7 

23.6074 

.40 

.8384 

.98 

3.2395 

3.8 

24.5710 

.42 

.9020 

LOO 

3.3390 

3.9 

25.5472 

.44 

.9717 

1.1 

3.8522 

4.0 

26.5360 

To  obviate  the  use  of  a  formula  so  encumbered, 
Table  2  has  been  computed.  In  which  the  1st,  3d 
and  5th  columns  represent  the  heads  or  depths  in 
feet,  from  the  level  of  still  water  to  the  crest;  the  2d, 


PRACTICAL   HYDRAULICS.  25 

4th  and  6th  columns  represent  the  quantities  dis- 
charged per  second,  for  the  given  depths  over  each 
lineal  foot  of  weir.  These  quantities  are  the  products 
of  the  unit  length,  cubes  of  the  square  roots  of  the 
given  depths,  and  their  respective  variable  coefficients 
found  in  Table  1.  If  Qt  denote  the  tabulated  quantity, 
so  discharged  per  lineal  foot,  ht,  the  given  head,  and 
cy,  the  variable  coefficient  due  that  head,  then  we  shall 
have  the  formula  by  which  Table  Q,  has  been  com- 
puted, viz.: 

et=e,fc*.  (29) 

Multiplying  (29)  by  (I  —  0.1  nh),  the  factor  for  cor- 
recting length  of  weir  as  in  (28),  and  putting  Q  equal 
discharge  over  a  weir  whose  length  is  three  or  more 
times  the  depth  of  water  on  crest,  and  there  results: 

Q=Qt  (l—0.1nh,)=:c,'(l—0.l  nh)h?  (30) 


WHENCE  TO  FIND  THE  DISCHARGE  OF  WATER  OVER  A 
WEIR  WITH  CORRECTIONS  MADE  DUE  VARIABLE  COEFFI- 
CIENT, AND  DEPTH  ON  CREST. 

Rule  12.  —  Deduct  from  the  given  length  of  the 
weir  one-tenth  the  depth  of  water  on  the  crest,  for 
each  and  every  end  contraction,  and  multiply  the 
length  so  corrected  by  the  quantity  in  "flow"  column 
opposite  the  given  head  in  Table  2. 

Rule  12  is  derived  from  Eq.  30  —  middle  right  hand 
member  employed.  The  extreme  right  hand  member 


26  PRACTICAL    HYDRAULICS. 

of  the  same  equation  expresses  the  value  .of  Q  in  terms 
of  the  corrected  length,  and  the  head  and  the  variable 
coefficient  as  found  in  Table  1. 

This,  in  fact,  is  the  modified  formula  of  (28),  by 
which  examples  16  and  17  were  solved. 

Ex.  18. — The  depth  of  water  being  two  feet  over  a 
crest  seven  feet  in  length,  what  is  the  discharge? 

Col.—  By  Rule  12. 

7— 2  X- 2=6. 6  feet,  corrected  length, 

BY  TABLE  2. 

9.4299  cubic  feet= discharge  due  head  of  two  feet. 
Whence,  9.4399x6.6=62.30  cubic  feet.— Ans. 
Cat. — By  formula  (30),  extreme  right  hand  member. 
7_2x.2=6.6  feet,  corrected  length. 

(2)t=2.8284=square  root  of  cube  of  depth. 

3.338=coefficient  by  Table  1,  due  head  of  two  feet. 

Q=3.338X6. 6X2.8284=62.32  cubic   feet.— 4ws. 

The  calculation  in  which  Table  2  is  employed,  is 
seen  to  be  much  shorter  and  far  more  simple  than 
that  in  which  Table  1  is  employed. 

Weirs  are  usually  constructed  with  horizontal  crests 
and  vertical  ends,  forming  a  notch,  through  which  the 
flow  of  water  is  measured  in  its  passage  from  a  reser- 
voir, or  other  storing  place;  or  in  its  passage  over  a 
submerged  dam  across  a  flume,  canal  or  natural 
stream.  A  weir  should  be  free  from  vibration.  Its 
crest  and  ends  should  be  chamfered  on  the  down- 
stream side  to  an  edge — say  one-tenth  of  an  inch 


PEACTICAL    HYDEAULICS.  27 

thick.  Its  upstream  face  should  be  vertical,  and  its 
downstream  face,  so  inclined  or  fashioned  as  not  to 
resist  the  flow  of  water.  On  the  upstream  side,  the 
depth  of  water  below  the  level  of  the  crest  should  be 
fully  twice  as  great  as  the  head  of  water  on  the  crest. 
The  head  or  depth  is  the  vertical  distance,  as  AC,  Fig. 
2,  between  the  crest  and  the  level  of  still  water  at  a 
point  some  distance  above  the  weir,  as  at  a.  In  ordi- 
nary practice,  the  head  is  measured  with  a  common 
rule  or  linear  measuring  scale,  from  the  top  of  a  post, 
P,  in  Fig.  2,  set  level  with  the  crest.  If  greater  accu- 
racy is  required,  the  "Boy den  Hook  Gauge"  should  be 
employed.  Care  must  be  taken  that  the  flow  over 
the  weir  shall  not  be  affected  by  the  approaching 
stream.  The  area  of  the  weir  opening  should  not 
exceed  a  fifth  part  of  that  of  the  supply  stream.  With 
proper  care  in  taking  the  data,  the  weir  affords  very 
accurate  means  of  measuring  the  flow  of  water. 

This,  taken  in  connection  with  the  weir's  simplicity 
and  facility  of  construction,  cheapness  and  wide  appli- 
cation, renders  it  of  great  practical  importance,  es- 
pecially to  those  concerned  in  the  measurement  of  flow- 
ing water,  in  places  of  difficult  access.  A  temporary 
dam,  in  illustration,  built  across  a  natural  stream,  with 
a  crest  board  firmly  fitted,  level  and  vertically  width- 
wise  to  the  dam's  top,  makes  a  good  measuring  weir. 
In  flumes  and  canals,  measuring  weirs  can  obviously 
be  constructed  with  equal  facility. 

The  weir  crest  being  about  three  feet  wide  and 
level,  with  a  rising  incline  to  its  receiving  edge,  Mr, 


28 


PRACTICAL  HYDRAULICS: 


Francis  offers  the  following  formula,  for  approximate 
measurements,  for  depths  between  six  and  eighteen 
inches: 

(2=3.01208  &/•»  (31) 

As  this  formula  is  somewhat  complicated,  the  writer 
would  present  Table  3,  computed  from  it,  which  will 
be  found  far  simpler  and  less  tedious  of  application. 


TABLE  3. 


Plow   for  given  depths  over  each  lineal  foot  of  weir,  with 
crest  three  feet  wide. 


Head. 

Flow 
Cubic  Feet 

Head. 

Flow 
Cubic  Feet 

Head. 

Flow 
Cubic  Feet 

6  inches. 
7.5     " 
9 

1.043 
1.467 
1.939 

10.5  inches 
12        " 
13.4     " 

2.445 
3.012 
3.607 

15.  inches. 
16.5     " 
18 

4.238 
4.903 
5.601 

The  depth  or  head  being  given. 

TO  FIND  THE  FLOW  OF  WATER  OVER    A    WEIR   WHOSE 
CREST  IS  THREE  FEET  WIDE. 


Rule  13. — From  "flow"  column,  opposite  the  give** 
head,  in  Table  3,  take  the  number  representing  the 
discharge  in  cubic  feet  over  one  lineal  foot,  which 
multiply  by  the  given  length  of  the  weir. 


PEACTICAL  HYDBAULICS.  29 

Ex.  19. — The  head  being  15  inches,  and  the  length 
of  weir,  whose  crest  is  three  feet,  being  ten  feet,  what 
is  the  approximate  discharge  in  cubic  feet  per  second? 

Gal. — In  Table  3,  in  "flow"  column,  opposite  15 
inches,  given  head,  find  4.238  cubic  feet. 

Whence,  4.238x10=42.38  cubic  feet.— Ans. 

TBIANGULAB  WEIBS. 

A  triangular  form  of  measuring  weir  has  been 
employed  with  favorable  results. 


To  determine  the  flow  of  water  through  a  triangu- 
lar weir: 

In  Fig.  3  let  ft~BC  represent  the  head  of  water  in 
feet,  from  the  apex  C  to  the  level,  AD,  of  still  water. 

p~—  given  ratio  between  the  head,  h,  and  the  width 
of  the  weir — that  is,  let  ph—AD,  EF,  or  any  width 
taken  at  pleasure. 

c= coefficient  of  discharge. 

Q= quantity  of  flow  in  cubic  feet. 

#=any  portion  of  the  head  h. 

2<7=force  of  gravity=32.2. 

v= velocity  due  (h — x). 

Then  in  general  by  (13)  will  v=c  (2g)?(h—x)    (32) 


30  PRACTICAL   HYDRAULICS. 


dQ=pc  (2g(h—xx  dx.  (33) 

Integrating   (33)  between  the  limits  of  a=0  and 
x=h: 

C=20  V  cubic  feet.  (34) 


QUADRANTAL   WEIR. 

In  a  quadrantal  weir,  that  is,  a  weir  in  which  the 
angle  ACD— 90°— a  right  angle— AD— 2EC=2h= 
ph;  hence,  p=2.  Making  c=.616;  (20)J==8.025. 

Substituting  these  values  of  pc  and  (2f/)3  in  (34). 

Q—2.6365  /J  per  second.  (35) 

If,  in  (35),  k"  representing  inches  be  substituted  for 
h,  which  represents  feet,  and  the  unit  of  time  be  made 
one  minute,  we  shall  have  the  following  formula, 
which  is  attributed  to  Professor  James  Thompson,  of 
Glasgow  University: 

Q=0. 317  h"*  per  minute.  (36) 

If  a  second  be  the  unit  of  time,  and   the  head  be 
in  inches,  we  have: 

Q^.  005385  h"*  per  second.  (37) 

Professor  Thompson  having  experimented  satisfac- 
torily with  the  quadrantal  weir,  pronounces  it  more 
simple  and  reliable  than  the  rectangular  weir,  in  that 
the  ratio  between  the  head  of  water  and  the  horizon- 
tal width  of  notch  is  constant;  in  that  the  flow  of 


PBACTICAL  HYDRAULICS.  31 

water  through  it  is  less  effected  by  the  "depth  from 
the  crest  to  the  bottom  of  the  channel  of  approach," 
and  finally,  in  that  the  coefficient  of  discharge  is  con- 
stant for  different  depths.  In  the  experiments  of 
Prof.  Thompson,  "the  volumes  of  water,"  says  J.  T. 
Fanning,  C.  E.,  "varied  from  .033  to  .6  cubic  feet  per 
second."  From  this  statement  it  is  deducible  that  the 
depths  of  water  varied  from  two  inches  to  6.6  inches 
(see  Table  4),  though  the  depths  given  by  Mr.  Fan- 
ning vary  from  two  to  four  inches. 

The  simplicity  of  this  weir,  its  cheapness,  and  the 
assurances  of  its  superiority,  as  stated,  have  induced 
the  computation  of  Table  4  f or  ^practical  use. 


PRACTICAL  HYDRAULICS. 


TABLE  4. 

Plow  of  Water  per  Second  over  a  Quadrant  Weir. 


Head 

Flow 

Head 

Flow 

Head 

Flow 

Head 

Flow 

Inch. 

Cub.  Feet 

[nch. 

Cub.  Feet 

Inch. 

Cub.  Feet 

Inch. 

Cub.  Feet 

1. 

.0053 

4. 

.1691 

7. 

.6852 

10. 

1.6713 

.1 

.0067 

4.1 

.1799 

7.1 

.7100 

10.1 

1.7134 

.2 

.0083 

4.2 

.1911 

7.2 

.7352 

10.2 

1.7562 

.3 

.0102 

4.3 

.2026 

7.3 

.7610 

10.3 

1.7995 

.4 

.0123 

4.4 

.2146 

7.4 

.7873 

10.4 

1.8435 

.5 

.0146 

4.5 

.2270 

7.5 

.8142 

10.5 

1.8882 

.6 

.0171 

4.6 

.2398 

7.6 

.8416 

10.6 

1.9334 

1.7 

.0199 

4.7 

.2531 

7.7 

.8696 

10.7 

1.9794 

1.8 

.0230 

4.8 

9668 

7.8 

.8981 

10.8 

20260 

1.9 

.0263 

4.9 

7^809 

7.9 

.9271 

10.9 

2.0732 

2. 

.0298 

5. 

.2954 

8. 

.9567 

11. 

2.1211 

2.1 

.0338 

5.1 

.3105 

1  8.1 

.9869 

11.1 

2.1696 

2.2 

.0379 

5.2 

.3259 

8.2 

1.0177 

11.2 

2.2187 

2.3 

.0424 

5.3 

.3418 

8.3 

1.0489 

11.3 

2.2686 

2.4 

.0472 

5.4 

.3593 

8.4 

1.0808 

11.4 

23192 

2.5 

.0522 

5.5 

.3750 

8.5 

1.1133 

11.5 

23703 

2.6 

.0576 

5.6 

.3922 

8.6 

1.1463 

11.6 

2.4222 

2.7 

.0633 

5.7 

.4100 

8.7 

1.1799 

11.7 

2.4748 

2.8 

.0693 

5.8 

.4282 

8.8 

1.2142 

11.8 

2.5280 

2.9 

.0757 

5.9 

.4468 

8.9 

1.2489 

11.9 

2.5818 

3. 

.0824 

6. 

.4661 

9. 

1.2843 

12. 

2.6365 

3.1 

.0894 

6.1 

.4857 

9.1 

1.3203 

13. 

3.2205 

3.2 

.0968 

6.2 

.5059 

9.2 

1.3568 

17. 

6.2979 

3.3 

.1046 

6.3 

.5266 

9.3 

1.3941 

19. 

8.3167 

3.4 

.1126 

6.4 

.5477 

9.4 

.4318 

23. 

13.4089 

3.5 

.1211 

6.5 

.5693 

9.5 

.4702 

29. 

23.9369 

3.6 

.1300 

6.6 

.5915 

9.6 

.5092 

31. 

28.2800 

3.7 

.1391 

6.7 

.6141 

9.7 

.5490 

37. 

44.0128 

3.8 

.1488 

6.8 

.6373 

9.8 

.5890 

41. 

56.8896 

3.9 

.1588 

6.9 

.6610 

9.9 

.6299 

43. 

64.0830 

47. 

80.0420 

Table  4  has  been  computed  from  formula  (37),  ID 
which  the  head  is  in  inches,  and  the  quantity  dis- 
charged per  second  is  in  cubic  feet,  The  tabulated 


PRACTICAL   HYDRAULICS.  33 

discharges  for  depths  of  water  over  six  inches  requires 
the  confirmation  of  experiment;  until  that  shall  be 
had,  however,  there  seems  good  reason  from  the  data 
to  regard  them  approximately  correct. 

APPLICATION  OF  TABLE  4. 

Ex.  20.— The  depth  of  water  in  a  quadrantal  weir 
being  2.1  inches,  what  is  the  discharge  over  it  in  cubic 
feet  per  second? 

Qalt — jn  "head"  column,  Table  4,  find  the  given 
head  2.1  inches,  opposite  which,  in  "flow"  column, 
will  be  found  .0338  cubic  feet:  the  quantity  sought. 


To  determine  the  flow  of  water  over  a  quadrantal 
weir,  the  head  given  being  equal  to  the  product  of 
two  factors,  each  designating  a  head  of  ivater  in 
Table  4. 

Rule  14. — Multiply  the  product  of  the  discharges 
due  the  factor  heads  by  189.2. 

Ex.  21. — The  head  in  a  quadrantal  weir  being  15 
inches,  what  is  the  discharge  per  second? 

Cat. — Observe  that  the  factors  of  the  given  head 
are  three  and  five:  3X5—15. 

By  Table  4,  flow  due  5"  head=.2954. 

By  Table  4,  flow  due  3"  head=.0824. 

Hence,  .2954 X. 0824X189. 2 =4. 6054  cubic  feet.— 
Ans, 


34  PRACTICAL  HYDRAULICS. 

Ex.  22. — The  head  in  a  quadrantal  weir  being  42 
inches,  what  is  the  discharge  per  second? 

Gal. — Observe  that  the  factors  of  the  given  head 
are  six  and  seven:  6x7=42  inches. 

By  Table  4,  flow  due  6"  head-. 4661. 

By  Table  4,  flow  due  7"  head=.6852. 

Hence,  .4661X.6852xl89.2=60.4258  cubicfeet.- 
A  ns. 

The  application  of  Rule  14  to  depths  below  13  inches 
will  be  unnecessary;  the  rule  is  given  to  avoid  an 
extended  table. 

EQUILATERAL  WEIR. 

To  determine  the  flow  of  water  over  an  equilateral 
weir,  EOF,  in  Figure  3,  .make  ^9  —  y.  That  is  when 
h  is  the  hight  of  an  equilateral  triangle,  its  side  or 
width  as  in  EOF,  is  EF=ph=  -.  Substituting  this 

1/3 

value  of  p  in  Eq.  (33J,  there  results: 

Q=  1.52217  /J"  per  second.  (38) 

Were  the  head  given  in  inches,  and  the  quantity  of 
flow  required  in  cubic  feet  per  minute,  then  would 

Q  =  1.77  h*  per  minute.  (39) 

The  heads  being  equal  in  the  two  forms  of  trian- 
gular weirs,  thus  far  considered,  then  will  the  dis- 
charge in  the  equilateral  form  be  to  the  discharge  in 
the  quadrantal  form  as  .57735  is  to  1.  Hence,  to  find 


PKACTICAL  HYDEAULICS.  35 

by  Table  4,  the  discharge  over  an  equilateral  weir, 
the  head  being  given: 

Rule  15. — Find  in  Table  4  the  discharge  due  the 
given  head  over  a  quadrantal  weir.  Multiply  the 
quantity  so  found  by  .57735. 

Ex.  23. — The  head  of  water  in  an  equilateral  weir 
is  eight  inches.  What  is  the  discharge  in  cubic  feet 
per  second? 

Gal.— By  Table  4,  it  is  seen  that  the  "flow"  due  the 
given  head,  eight  inches,  is  .9567  cubic  feet  per  second. 
Hence,  .9567 X. 57735 =.5524  cubic  feet.— Ans. 

In  Eq.  (34)  any  value  may  be  given  p,  as  1,  2,  3,  4, 
5,  etc.,  so  as  to  indicate  the  relations  existing  between 
the  hight  and  width  of  triangular  weirs.  But  since 
c,  the  coefficient  of  discharge,  varies  with  every  differ- 
ent condition  imposed,  the  labor  of  determining  the 
theoretical  flow  due  any  considerable  number  of  such 
forms  would  necessarily  be  barren  of  practical  results. 

In  general  let  (34)  be  reduced  to  its  simplest  form: 

Q^Z.Upch?.  (40) 

In  which  h  denotes  feet,    nd  Q  cubic  feet. 

If  in  (40)  we  make  p=2  ^ind  c=.616,  the  formula 
becomes  that  of  the  quadrantal  weir,  as  shown  by 
Eq.  (35). 

If  we  make  p =  -r=1.1547,  and  c  =  .6!6,  the  for- 
mula becomes  that  of  the  equilateral  weir,  as  shown 
byEq.  (38.) 

If  we  make  j>=2]/3^3.4641,  and  c  =  .616,  the  for- 


36  PRACTICAL  HYDRAULICS. 

mula  becomes  that  of  a  weir,  whose  apex  or  angle 
C=120°;  Fig.  3. 

Q-4.5665  h*.  (41) 

And  in  this  manner  may  special  formulas  be  de- 
duced from  (40),  to  meet  the  various  requirements  of 
triangular  weirs.  In  theory  the  results  obtained  are 
as  they  should  be ;  but  in  fact,  experiment  best  de- 
termines in  what  cases  c  is  constant,  and  in  what 
variable. 

TRAPEZOIDAL  WEIRS. 

To  determine  the  flow  of  water  over  a  trapezoidal 
weir,  let,  in  Fig.  3,  ACD  represent  a  triangular 
weir,  in  which  GI  is  a  line  indicating  a  division  with 
respect  to  flow  of  water  over  the  weir. 

The  flow  of  the  triangular  portion,  GCI,  taken 
from  the  entire  flow  due  ACD,  there  remains  that 
portion  of  the  flow  due  the  trapezoid  DAGI.  The 
mean  velocity  of  the  water  in  ACD  is'  evidently 
greater  than  the  mean  velocity  is  in  DAGI,  else  the 
flow  in  the  trapezoid  could  readily  be  determined  from 
that  of  ACD  by  the  simple  ratio  of  these  areas. 

Let,  in  the  present  solution,  h,  p,  c,  Q,  g  and  v  have 
the  same  values  or  functions  which  they  had  in  the 
discussion  hitherto  of  triangular  weirs.  In  addition 
put  CL=?i/i; 

Then  BL^  (I— n)  h,  (42) 

the  depth  of  the  trapezoid  DAGI. 

Integrating  equation  (33)  between  the  limits  of  a?=0 


PRACTICAL   HYDRAULICS.  37 

and  x—nh,  and  there  results  the  flow  per  second  due 
the  triangular  portion  of  the  weir  GCI. 


Q=pc  (2«/)-|  (l-«)+|  (1—  n)+  ,«,&.        (43) 
Deducting  (43)  from  (34),  there  remains: 

-Ti)lAi.  (44) 


Formula  (44)  represents  the  discharge  due  the 
brapezoid  DAGI. 

y  -v  «.  * 

(44) 

Ex.  24. — The  width  of  a  trapezoidal  weir  being 
bwo  feet,  the  depth  one-fourth  (J)  of  a  foot,  the  sides 
inclining  45°  to  the  horizon,  and  the  coefficient  of  dis- 
charge being  .62,  required  the  cubic  feet  flow  over  it 
per  second? 

Gal. — Since  the  width  is  two  feet,  and  the  inclina- 
tion of  the  sides  45°,  were  the  sides  produced  down- 
ward till  they  meet,  the  depth  of  the  triangle  so  formed 
would  be  equal  to  one-half  the  given  width — that  is, 
k—\  foot,  and  j9=2. 

The  given  depth  of  weir  being  J,  hence,  n—(~L — J) 

__  3 

Making  substitution  of  these  values  in  (44), 
2  +  3x1=4.25; 


38  PRACTICAL    HYDRAULICS. 

(1)  *=1.     We  have 
Q=1.07x2X.62xl25Xi=7.05  cubic  feet.— Ans. 

Ex.  25. — The  width  of  a  trapezoidal  weir  being  4.5 
feet,  the  depth  one  foot,  the  sides  inclining  45°  to  the 
horizon,  and  the  coefficient  of  discharge  being  .62,  re- 
quired the  cubic  feet  flow  over  it  per  second. 

Gal.  1st. — The  width  being  4.5  feet,  and  the  incli- 
nation of  the  sides  45°,  if  the  sides  be  produced  till 
they  meet,  the  depth  of  the  triangle  so  formed  will  be 
2.25=,£  feet. 

2.25—1-1.25;  7i^f|-=f;  l—w  =  l— f  =  {;  p=_2. 

Substituting  these  values  in  formula  (44), 

Q=  1.07X2X6.2  (2+V)  (J)*(£)*=10.946  cubic 
feet. — Ans. 

Gal.  2d.— The  bottom  width  1.25X2=2.5  feet. 

Regard  the  trapezoidal  weir  made  up  of  a  rectangular 
weir,  whose  length  is  2.5  feet  and  depth  one  foot,  and 
of  a  quadrantal  weir,  whose  width  is  two  feet  and 
depth  one  foot.  Then: 

The  quantities  of  "flow"  are  found  to  be  by  Table  2, 
for  each  linear  foot  of  crest,.  3.339  cubic  feet;  hence, 
for  2.5  feet,  3.339x2.5—8.3475  cubic  feet,  and  by 
Table  4,  for  "flow"  over  a  quadrantal  weir  one  foot 
deep=2.6365  cubic  feet. 

Hence,  8.3475  +  2.6365=10.984  cubic  feet.— Ans. 

The  discrepancy  between  the  results  obtained  by 
the  1st  and  2d  calculations  arises  from  the  different 
values  assigned  the  coefficients  of  discharge. 


PRACTICAL    HYDRAULICS.  39 

By  Gal.  1st  the  coefficient  was  taken,  as  proposed  in 
the  given  problem,  at  .62. 

By  Cal.  2d,  the  coefficient  employed  in  computing 
Table  2,  was  taken  from  Table  1,  which  will  be  seen 
to  be  3.339.  So  that  the  coefficient  of  discharge  em- 
ployed in  Table  2,  was,  in  fact,  3.39-s-J  (8.025)= 
.6241,  instead  of  .62,  as  provided  in  the  given  propo- 
sition. The  coefficient  of  discharge  employed  in  com- 
puting Table  4,  was  .616,  which  was  deduced  from 
the  formula  given  by  Prof.  Thompson.  Weisbach's 
Mechanics  and  Engineering  state  that  the  coefficient 
employed  by  Prof.  Thompson,  was  .619,  while  J.  W. 
Stone,  C.  E.,  author  of  Hydraulic  Formula,  states 
that  it  was  .617.  • 

Cal.  3d.—  The  top  width  is  given  4.5  feet.  The 
bottom  width=(2.25—  l)X2=-2.5  feet.  Mean  width 


By  Francis'  formula  (3.5—  .1X2X1)^3.3,  cor- 
rected length. 

By  Table  2,  flow  over  each  linear  foot,  3.339  cubic 
feet.  Hence,  3.339X3.3=11.008  cubic  feet.—  Ans. 

This  result  differs  but  little  from  those  obtained 
from  more  rigorous  solutions. 

FLOW  OF  WATER  OVER  TRAPEZOIDAL  WEIRS  IN  WHICH 
THE  LENGTH  OF  THE  CREST  IS  GREATER  THAN  THE  TOP 
WIDTH  OF  THE  NOTCH. 

Let   ABCD,   Fig.  4,    represent  a  trapezoidal  weir 


40  PRACTICAL   HYDKAULICS. 

in  which   the  length  of  the  crest  AB=&  is  greater 
than  the  width  of  the  water  surface  CD=t. 


FIG.  4. 

Draw  CE  parallel  to  DB,  and  let  fall  on  BA,  the 
perpendicular  DG,  CF.  Then  will  the  rhomboid, 
ECDB,  equal  in  area  the  parallelogram  FCDG,  and 
the  entire  weir,  ACDB,  equal  in  area  FCDG  +  ACE. 

Let  h—thG  head  or  depth  of  water  between  the 
levels  of  the  crest  and  still  water;  p=the  ratio  of  the 
base,  EA,  to  hight,  FC,  of  the  triangle,  ACE;  c=  co- 
efficient of  flow;  #=any  head  not  greater  than  h. 

Then  dQ=c  (Zg)~*(t  +  px)  x^dx.  (45) 

Integrating  (45),  observing  that  Q—Q, 
When  05=0, 

Q=c  (2g)*(^t  A*  +  ^>  h%).  (46) 

Reducing  (46),  observing  that  6=3  (t+ph), 

(47) 


Making  b—ph,  £=0,  that  is,  closing  the  top  of  the 
weir,  at  the  level  of  still  water,  and  equation  (46) 
becomes 


cubic  feet.  (48) 

&0 

Comparing  equations  (48)  and  (34), 


PRACTICAL  HYDRAULICS,  41 

•0:0 ::!:!::  8:  2.  (49) 

48    34      15    15 

Equation  (49J  shows  that,  with  respect  to  two  tri- 
angular weirs  of  equal  size,  the  discharging  ca- 
pacity of  the  weir  whose  top  is  closed  at  the  level  of 


still  water,  as  E  in  Fig.  6,  is,  to  the  discharging  ca- 
pacity of  the  weir  whose  top  is  open,  as  AB  in  Fig.  5, 
as  3  is  to  2,  or  1.5: 1.  Table  4  was  computed  from  an 
open  weir,  as  ABC,  Fig.  5.  To  render  it  applicable 
to  a  closed  weir,  as  DEF : 

Rule  16. — Multiply  the  tabulated  flow  due  any 
head  given  in  Table  4  by  1.5. 

Ex.  26. — In  a  trapezoidal  weir,  the  head  between 
the  crest  and  the  level  of  still  water  being  three  (3) 
inches,  the  length  of  the  crest  two  (2)  feet,  and  the 
width  of  the  opening  at  water  level  one  and  three- 
fourths  (1.75)  feet,  what  is  the  flow  in  cubic  feet  per 
second  when  the  coefficient  of  discharge  is  .62  ? 

Gal. — Head  3  inches=J  feet. 

By  formula  (47),  (J)*=J. 

Three  times  bottom  width:   2x3=6. 

Twice  top  width:    1.75x2=3.5. 

Whence,  1.07X-62  (3.5+6)  £=.7878  cubic  feet, - 

Ans. 


42  PRACTICAL   HYDRAULICS. 

Ex.  27. — In  a  trapezoidal  weir,  the  head  being 
twelve  inches,  the  bottom  width  three  feet,  the  top 
width  one  foot,  and  the  coefficient  of  discharge  .624, 
what  is  the  flow  of  water  over  it  in  cubic  feet  per  sec- 
ond ?  By  formula  (47). 

Cal.lst.— LOT X. 62  (1X2+3X3)  Xl=T.'3445  cu- 
bic feet. — Ans. 

Cat.  2d. — Observe  that  the  weir  opening,  ACDB, 
Fig.  4,  is  resolvable  into  two  parts,  to  wit:  the  part 
CDBE,  which  is  equal  to  the  rectangle  CDGF, 
and  the  triangular  part  ACE,  whose  crest  is  AE, 
and  whose  top  is  closed  at  C,  at  the  level  of  still  water. 
Applying,  in  Example  (27),  Table  2,  to  the  rectangular 
part  CDGF,  substituted  for  the  part  CDBE,  and 
Table  4,  to  the  triangular  part  ACE: 

By  Table  2,  due  one  foot  head,  one  foot  crest,  3.3390 
cubic  feet. 

By  Table  4,  due  in  quadrantal  weir  with  open  top, 
12-inch  head,  2.6365. 

By  Rule  16,  2.6365x1.5=3.9547. 

Hence  3.3390+3.9547=7.2937  cubic  feet.— Ans. 

The  discrepancy  between  calculating  1st  and  2d 
arises  from  Table  4,  in  the  computation  of  which 
.616  on  the  authority  of  Prof.  Thompson,  was  em- 
ployed as  the  coefficient  of  discharge,  instead  of  .624, 
as  proposed  in  the  given  example. 


PEACTICAL  HYDRAULICS.  43 


FLOW  OF  WATEK  OVER  A  RECTANGULAR  WEIR,  HAV- 
ING ITS  ANGLES,  HORIZONTAL  AND  VERTICAL,  AND  ITS 
UPPERMOST  ANGLE  ON  THE  LEVEL  OF  STILL  WATER. 


Let  ABCD,  of  Fig.  7,  represent  a  rectangular 
weir,  having  its  vertical  angle  C,  at  the  level  of  still 
water.  Through  C,  draw  a  horizontal  line  indefinitely. 
Produce  AB,  AD,  intersecting  this  horizontal  line 
in  fi  and  F.  Draw  A.G=h  perpendicularly  to  EF, 
and  bisecting  A. 

Let  AB=m,  and  AD=7i.  It  is  obvious  from  the  im- 
posed conditions,  that  AE=AF=m+7i;  that  EF= 
2GE=2GF=2AG:=2X-  that  FD=DC=AB=m, 
and  BE=BC=AD=™. 

Observe  that  in  Fig.  7  are  represented  three  open 
quadrantal  weirs,  FAE,  FDC  and  CBE,  and  that  the 
given  rectangle  ABCD-=FAE— FDG— -CBE.  (50) 

Denote  by  Qu  the  flow  of  the  given  rectangular 
weir. 

By  similar  triangles,  find  the  heads  or  depths  as 
follows; 

(    mh   ) 

PD=  \—r\ 

\m-\-n ) 


44  PEACTICAL    HYDRAULICS. 


(    nh    ) 
LB=  (52) 


Substituting  these  values,  and  the  value  of  A.G~h 
in  (34),  noting  that  for  quadrantal  weirs,  p=2, 


(53) 


If  in  (53),  the  general  formula  for  the  flow  of  water 
through  a  rectangular  weir  having  its  uppermost  an- 
gle vertical  at  the  level  of  still  water,  we  make  m 
equal  n,  and  substitute  the  value  of  (2^)2  =8.025,  and 
c  =  .616, 

^=1.70435^*.  (54) 

In  which  case  the  rectangle  becomes  a  square,  as 
represented  in  Fig.  7,  by  AIGN. 

Comparing  formula  (54)  with  formula  (35),  which 
is  for  the  flow  of  water  over  a  quad  ran  tal  weir,  we 
have: 

1.70435 


To  determine  the  flow  of  water  through  a  square 
weir,  having  its  uppermost  vertical  angle  at  the  level 
of  still  water. 

Rule  17.  —  According  to  formula  (54),  multiply  the 
square  root  of  the  fifth  power  of  the  head  or  depth  by 
1.70435;  or  by  formula  (55),  multiply  the  flow  in 
Table  4  for  the  given  head  by  .6464. 


PRACTICAL   HYDRAULICS.  45 

Ex.  27.  —  The  head  being  3  inches=  J  foot  in  a  rec- 
tangular weir,  having  its  upper  most  vertical  angle  at 
level  of  still  water,  what  is  the  flow  in  cubic  feet  per 
second? 

Cal.  1st.—  By  formula  (54). 

Fifth  power  of  the  square  root  of  (J)"2'  =  jV 

1.70435XA=--05826  cubic  feet.=4m 

Cal.  2d  —  By  Rule  17,  second  part. 

By  Table  4,  flow  due  3-inch  head  =.0824.  Then 
.0824  X.  6464=.  05326  cubic  feet.—  Ana. 

Ex.  28.  —  The  sides  of  a  rectangular  weir,  with  its 
angles  vertical  and  horizontal  being  2  feet  and  1  foot, 
the  coefficient  of  discharge  being  .62,  what  is  the  flow 
per  second? 

Cal.  —  Employ  formula  (53). 

Taking  the  given  data  m=2,  n=l> 

Then  m  +  fi™  3,  and  (see  Fig.  7), 


6.55376. 

By  Table  5,  ml=(2)^ 

By  Table  5,  (m+w)*  =  (3)^=15.590. 

Substituting  these    values    of    m"2,    ri*t 
/J,  (2#)i=8.025,  and  c=.62.       Q,  =T8yX-  62x8.  025 


1559 

Whence,  Q/y-9.9644  cubic  feet.—  Anst 


46  PRACTICAL   HYDRAULICS. 


FLOW   OF    WATER    THROUGH    CIRCULAR    AND    SEMI- 
CIRCULAR WEIRS. 


Let  Figs.  8,  9  and  10,  represent  respectively  the 
circular  and  semi-circular  weirs,  Fig.  8  touching  the 
water  surface  at  A,  Fig.  9  in  a  similar  manner  at  B, 
and  Fig.  10  at  its  diameter  CD. 

Let  r  feet  denote  the  radius  with  which  the  several 
weirs  are  described,  then  in  both  Figs.  9  and  10  will  r 
denote  the  head,  while  in  Fig.  8  the  head  (maximum) 
will  be  2r. 

Let  x  in  Fig.  8  denote  any  portion  of  the  head,  and 

Qt  the  flow  in  cubic  feet,  (2#)"2'  acceleration    of    grav- 
ity, and  c  coefficient  of  discharge. 


Then   dQ=2c(2g  *  2  (r)i    (l—ffixdx.        (56) 

Integrating  (56)  between  limits  of  oj-=0,  and  x=2r, 
and  substituting  the  value  of  (20)i=8.025t 

Q,=24.2129  crl  (57) 


PRACTICAL   HYDRAULICS.  47 

Let  Q2—ttie  flow  in  Fig.  9  per  second.  Integrating 
(56)  between  limits  of  x=0,  and  x=r,  there  results  the 
flow  in  that  portion  of  Fig.  8  represented  by  FAE, 
which  is  equal  to  HBG,  Fig.  9,  the  discharge  sought, 
viz.: 


(58) 

Again  in  Fig.  10  :  Let  x  denote  any  portion  of  the 
head  from  A,  and  Q3  the  flow  in  cubic  feet  per  second  ; 
Then 

d  Qs=  (20)*  (r  —  x*fix%da>.  (5  9) 

Integrating  (59)  between  limits  #=0,  and  x=r,  and 
substituting  value  of  (2#)i=8.025, 

#3-7.6932  crl  (60) 

Comparing  equations  (60)  and  (35)  and  making 
c=.616,  and  r=/i, 

(61) 


To  find  by  Table  4  the  flow  through  a  semi-circular 
weir:  Open  at  the  top  as  represented  by  CD  Fig.  10. 

Rule  18.—  Multiply  the  flow  in  Table  4  for  the 
given  head  or  radius  by  1.79.  See  formula  (61). 

The  triangle  CA2D,  inscribed  in  the  semi-circular 
weir,  Fig.  10,  represents  a  quadrantal  weir  whose 
flow  is  Q,  while  the  flow  of  the  semi-circular  weir 
CA2D  is  Q3. 

Comparing  equations  (58)  and  (35)  and  making 
c=.616,  and  r=h, 

(62) 


48  PRACTICAL  HYDRAULICS. 

To  find  by  Table  4,  the  flow  through  a  semi-circular 
weir  closed  at  the  top,  as  represented  at  B  Fig.  9. 

Rule  19.— Multiply  the  flow  in  Table  4  for  the 
given  head  or  radius  by  2.1568  (62). 

To  find  by  Table  4  the  flow  through  a  circular 
weir  touching  the  water  surface  at  A  as  represented 
in  Fig.  8. 

Comparing  equations  (57)  and  (35)  and  making 
c=.616,  andr=fc, 

$,=5.6566  Q.  (63) 

Rule  20.— Multiply  the  flow  in  Table  4  for  the  head 
or  depth  equal  to  the  given  radius  of  the  circular  weir 
or  opening  by  5.6566.  See  formula  (63). 

Ex  29. — In  a  semi-circular  weir,  with  open  top,  as 
represented  by  Fig.  10,  the  head  or  radius  is  ten  inches. 
What  is  the  discharge  in  cubic  feet  per  second? 

Cal. — By  Table  4  the  flow  due  a  head  of  10  inches 
is  1.6713  cubic  feet. 

By  Rule  18  we  have — 

1.6713X1.79  =  2.9916  cubic  feet.— Ans. 

Ex.  30. — In  a  semi-circular  weir  or  opening  with 
closed  top,  as  represented  by  Fig.  9,  the  head  or  radius 
being  ten  inches,  what  is  the  flow  in  cubic  feet  per 
second  ? 

Gal. — By  Table  4  the  flow  due  a  head  of  ten  inches 
is  1.6713  cubic  feet. 

By  Rule  19  there  results — 

1.6713X2.1568=3.6047  cubic  feet.— Ans. 

Ex.  31. — In  a  circular  weir  or  opening  touching  the 


PBACTICAL  HYDRAULICS. 


49 


water  surface,  as  at  A,  Fig.  8,  the  radius  is  eight  inches, 
required  the  cubic  feet  flow  per  second. 

Gal.  1st. — By  Table  4  the  flow  due  a  head  of  eight 
inches  is  .9567  cubic  feet. 

By  Rule  20  we  have — 

.9567X56.566=-5:4117  cubic  feet. — Ans. 

Gal.  2d. — By  formula  (57)  :     8  inches^  f  feet. 

By  Table  5,   (f)*=^=. 3629  nearly: 
24.2129X.616X.3629=5.4117  cubic  feet.— Ans. 

FLOW  OF  WATER  THROUGH  PARABOLIC  WEIRS. 


Let  Figs.  11  and  12  represent  parabolic  weirs,  touch- 
ing the  water  surface,  Fig.  11,  at  its  apex,  A,  and  Fig. 
12  at  its  inverted  base,  EF. 

Let  h,  in  each  weir,  as  AD  or  HA,  denote  the  head, 
and  let  b  denote  the  base,  as  BD,  DO  or  EH,  FH. 

Let  c=the  coefficient  of  discharge. 

Q4= the  quantity  discharged  in  cubic  feet  per  second, 
and  (2#)i  -=8.025  due  gravity. 

Let  a^any  part  of  the  head,  estimated  from  A,  in 
Fig.  11. 


50  PRACTICAL   HYDBAULICS. 

The  equation  of  the  parabola,  in  which  x  and  y  are 
co-ordinates,  is: 


y*-=2px;  whence  y  =  (2px.  (64) 

The  equation  of  flow  is: 

dQ4=c  (20)i(2p)iadte.  '  (65) 


Integrating  (65)  between  limits   x=Q,   and    x=h; 
and  substituting  the  values  of  (2jo)i=p,  (2#)i  -8.025; 


(66) 
Let  Q5—  the  flow  in  weir  represented  by  Fig.  12. 

(67) 


Integrating  (67)  between  limits  o?=0,  and  #—  &,  and 
substituting  the  values  of  (2p)^=      and  (2#)i=8.025  ; 


•Q5-3.1546  c  b  h*.  (68) 

Assume  any  ratio,  n,  to  exist  between  the  base  b, 
and  the  hight  or  head,  h: 

As  b=nh.  (69) 

Then  Q5=-3.1546  w  h*.  (70) 

This  formula  is  adapted  to  the  finding  of  the  flow 
of  water  over  both  shallow  and  deep  weirs.     Thus  by 


PBACTICAL  HYDRAULICS.  51 

making  n  successively  equal  to  1,  2,  3,  4,  5,  6,  7,  etc., 
the  represented  flow  in  (76)  becomes  correspondingly 
affected.  To  accomplish  a  similar  result  by  the  semi- 
circular weir,  would  be  no  easy  task,  requiring  the 
employment  of  a  very  intricate  and  unwieldy  formula 
or  extensive  table. 

Making  in  Eq.  70,  n=2,  there  results: 

g5=r6.3092cfci  (71) 

In  this  case,  b—Zk,  and  hence: 

Comparing  equations  (71)  and  (35),  and  making  c 
=  .616, 

Q5=1.4734e.  (72) 


To  FIND,  BY  TABLE  4,  THE  FLOW  OF  WATER  OVER  A 

PARABOLIC  WEIR,  WITH  AN  OPEN  TOP  —  THE  WIDTH  BE- 
ING EQUAL  TO  TWICE  THE  DEPTH  OR  HEAD. 


Rule  21.— Multiply  the  flow  in  Table  4  for  the  head 
or  depth,  equal  the  given  head,  by  1.4734.  See  for- 
mula (72). 

Ex.  32. — In  a  parabolic  weir,  with  open  top,  the 
head  is  11  inches,  and  the  width  22  inches.  What  is 
the  discharge  of  water  through  it  in  cubic  feet  per 
second  ? 

Gal.  by  Table  4. — Flow  corresponding  to  head  of  11 
inches,  2.1696  cubic  feet. 
Then  by  Rule  21: 


52  PRACTICAL    HYDRAULICS. 

2.1696X1.4734=3.1967  cubic  te&t.—Ans. 
Comparing  equations  (70J  and  (35),   and  making 
c-:.6l6, 

(73) 


To  FIND,  BY  TABLE  4,  THE  FLOW  OF  WATER  OVER  A 

PARABOLIC  WEIR,  WITH  OPEN  TOP  —  THE  WIDTH  BEING  A 
GIVEN  NUMBER,  n  TIMES,  THE  HEAD  OR  DEPTH. 

Rule  22.—  Multiply  the  flow  in  Table  4,  for  the 
head  or  depth,  equal  to  the  given  head,  by  .73705 
times  the  ratio  between  the  given  head  and  width. 
See  formula  (73). 

Ex.  33.  —  In  an  open  parabolic  weir,  the  head  being 
10  inches,  and  the  width  50  inches  —  that  is,  5  times  10 
inches  (71=0),  required  the  cubic  feet  flow  per  second. 

Cal—By  Table  4,  flow  due  10  inches,  1.6713. 

By  Rule  22.—  1.6713X  .73705X  5=6.1592  cubic 
feet.  —  Ans. 

Comparing  equations  (66)  and  (35),  making  b=nh, 
as  in  (69),  and  c=.  616, 

(74) 


TO  FIND  THE  FLOW  OF  WATER  THROUGH  A  PARABOLIC 
WEIR  WHOSE  APEX,  A,  FlG.  11,  IS  AT  THE  LEVEL  OF 
STILL  WATER. 

Rule  23.  —  Multiply  the  flow  in  Table  4,  due  the 
head  or  depth  equal  to  the  given  head,  by  .9375  times 
the  ratio,  n,  between  the  given  head  and  width. 


PRACTICAL   HYDRAULICS.  53 

Ex.  34. — In  a  parabolic  opening  or  weir,  whose 
apex  reaches  the  surface  of  still  water,  the  head  or 
depth  being  23  inches,  and  the  width  230  inches — 
that  is,  10  times  23  inches  (n—~LQ),  required  the  flow 
in  cubic  feet  per  second. 

Col.— By  Table  4,  flow  due  23  inches,  13.4089. 

By  Rule  23,  13.4089X  .9375X  10  =125.71  cubic 
feet. — Ans. 


FLOW  OF  WATER  THROUGH  A  SUBMERGED  TRIANGULAR 
OPENING,  HAVING  ITS  VERTEX  BELOW    TE  BASE. 


Let,  in  Fig.  13,  BCD  represent  the  opening,  in 
which  6=BD,  the  width  at  top;  h=  AE,  head  on  the 
top  of  orifice;  ht—  AC,  head  on  its  bottom;  c=coefti- 

cient  of  discharge;  (2^f)^=acceleration  of  gravity; 
Q=discharge  in  cubic  feet  per  second;  &=any  part  of 
EC,  estimated  from  E;  a=EC,  depth  of  opening. 


Then  dQ=c  (2g)*(h+xdx.  (75) 


54  PRACTICAL  HYDRAULICS. 

Integrating  (75)  between  the  limits  x=Q,  and  x— 
h,—h, 


If  in  (76),  we  make  h=Q,  and  b=2a,  there  results: 

*.  (77) 


Equation  (77),  derived  from  the  general  equation 
(76),  is  seen  to  be  identical  with  equation  (35),  for 
the  flow  of  water  over  a  quadrantal  weir. 

In  equation  (76),  denote  the  ratio  between  the  head 
on  the  bottom  and  the  head  on  the  top  of  the  triangu- 

lar  opening   by   m;   thus    -=m,   and  substitute  the 

h, 

value  of  (2#)i=8.025. 

l  (78J 


To  FIND  THE  FLOW  OF  WATER  THROUGH  A  SUBMERGED 
TRIANGULAR  ORIFICE,  HAVING  ITS  VERTEX  BELOW  THE 
BASE. 


Rule  24.— From  J,  subtract  \  of  the  ratio  of  the 
given  heads  on  the  bottom  and  top  of  the  orifice,  and 
multiply  this  difference  by  the  cube  of  the  square  root 
of  the  same  ratio;  subtract  the  product  from  T2T; 


PBACTIOAL  HYDKAULICS.  55 

multiply  the  remainder  by  16.05  times  the  product  of 
the  ratio  between  the  depth  and  width  of  the  orifice, 
the  fifth  power  of  the  square  root  of  the  head  on  the 
bottom,  and  the  coefficient  of  discharge. 

Ex.  35.  —  In  a  submerged  triangular  orifice,  repre- 
sented by  Fig.  13,  the  head,  AC,  on  the  bottom  =h  = 
2.25  feet;  the  head,  AE,  on  the  iop=h=l  foot;  the 
width,  BD=6=5  feet;  the  depth  EC=a=1.25  feet; 
and  the  coefficient  of  discharge  c—  .616.  What  is  the 
flow  in  cubic  feet  per  second? 

Gal  1st.—  By  formula  (78)  and  Rule  24,  derived 
therefrom  : 

Ratio  of  heads,  ^=m=-s^v=%', 

Difference  (H?)=U; 

Cube  of  square  root  of  ratio,  m~2—  (f)"2—  /T. 

Product  of  this  difference,  and  the  cube  of  the 
square  root  of  the  ratio  of  the  given  heads, 


—  ITTT- 
Difference,  £-^1^%- 

Q=16.05X  .616X  T.VT  X  ^  X  W  =  18-29  cubic 
feet.  —  Ans. 

Gal.  2d.  —  Assuming  that  the  effective  head  is  the 
mean  of  the  given  heads  on  the  top  and  bottom  of  the 
orifice,  then  will  the  velocity  be  as  per  equation  (25) 
or  Rule  (8). 

^.616X8.025  (*i+*)i=(ii»|±i)i=6.3  feet  per  sec- 
ond. 

Area  orifice=5X  1.25-^2=3.125  square  feet. 


56  PRACTICAL    HYDBAULICS. 

Discharge  equal  to  the  product  of  the  velocity  and 
the  area  of  the  orifice,  Q=6.3x 3.125=19.69  cubic 
feet  per  second. 

Gal.  3. — Assume  that  the  true  head  is  on  the  center 
of  gravity  of  the  opening  geometrically  considered, 
la  a  triangle,  the  center  of  gravity  is  at  the  inter- 
section of  right  lines  drawn  from  any  two  angles  and 
Usecting  the  opposite  sides.  Its  distance,  estimated, 
from  an  angle,  is  equal  to  two-thirds  the  length  of  the 
bisecting  line;  or  estimated  from  the  middle  of  a  side, 
is  equal  to  one-third  the  length  of  the  bisecting  line. 

In  Fig.  13,  AG=//=£+f=l  +  i-|l==1.4167. 

By  formula  (13),  modified  by  coefficient  ^=.616= 
8.025  (1.4167)i=5.8836  feet  per  second,  area  of  ori- 
fice, 5X1.25-^2=3.125  square  feet. 

Discharge  equal  to  the  product  of  the  velocity  and 
the  area  of  orifice,  Q=5.8836X3.125=18.39  cubic 
feet  per  second. 

Comparing  these  results,  it  is  seen 'that  the  second  is 
seven  and  six-tenths  per  cent  (.076)  too  great,  and  the 
third  fifty-four  one  hundredth  of  one  per  cent  (.0054) 
too  great. 

The  rule  generally  adopted  is,  that  "in  all  cases  when 
the  center  of  gravity  of  an  orifice  lies  at  least  as  deep 
below  the  fluid  surface  as  the  figure  is  high,"  the 
depth  ht  (that  is,  the  depth  at  the  center  of  gravity), 
of  this  point  may  be  regarded  the  head  of  water. 
This  rule  may  approximate  the  truth  sufficiently  close 
for  ordinary  practice,  but  is  not  to  be  employed  when 
a  high  degree  of  accuracy  is  required. 


PRACTICAL  HYDRAULICS, 


57 


FLOW  OF  WATER  THROUGH  A  SUBMERGED  TRIANGULAR 
ORIFICE  HAVING  ITS  VERTEX  ABOVE  THE  BASE. 


Let,  in  Fig.  14,  BCD  represent  the  opening,  in 
which  fr=BD,  the  width  at  bottom;  h=A.C,  head  or 
vertex;  hJ=A'E,  head  or  bottom;  a=EC,  depth  of 

opening;  (2(/^)=acceleration  of  gravity;  c=coefficient 
of  discharge;  Q— flow  in  cubic  feet  per  second;  x= 
any  part  of  a— EC,  estimated  from  C : 


(79) 


Then  dQ=c-    -(h+x)*x  dx. 


Integrating  (79J  between  the  limits  x=0,  and  x=a 


(80) 


Denote  the  ratio  between  the  head  on  the  bottom 
and  that  on  the  vertex  by  m: 

h 

m=-;  h=mhr 


58  PRACTICAL  HYDRAULICS. 

Substitute  the  value  of  h  in  (80),  and  the  value  of 


(81) 


a         .      15 


If  in  the  general  equation  (80),  we  make  h=0,  and 
b  =2  a,  there  results: 

Q=*c(2g)*h?,  (82) 

which  is  identical  with  formula  (48)  for  the  flow  of 
water  through  a  quadrantal  weir  having  its  apex  at 
the  water's  surface. 


To  FIND  THE  FLOW  OF  WATER  THROUGH  A  SUBMERGED 
TRIANGULAR  ORIFICE  HAVING  ITS  VERTEX  ABOVE  THE 
BASE. 


Rule  25. — From  £  subtract  J  of  the  ratio  of  the 
head  on  the  bottom  to  that  on  the  apex  of  the  orifice ; 
add  this  difference  to  the  T27  part  of  the  fifth  power  of 
the  square  root  of  the  same  ratio;  multiply  this  sum 
by  16.. 05  times  the  product  of  the  ratio  between  the 
depth  and  width  of  the  orifice,  and  the  fifth  power  of 
the  square  root  of  the  head  on  the  bottom.  Derived 
from  formula  (81). 

Ex.  36. — In  a  submerged  triangular  orifice,  repre- 
sented by  Fig.  14,  the  head,  AC,  on  the  apex=/i=l 


PBACTICAL    HYDRAULICS.  59 

foot  ;  the  head,  AE,  on  the  bottom=&/—  2.25  ;  the 
width,  BD=6=5  feet;  the  depth,.  CE=a=1.25;  and 
the  coefficient  of  discharge,  c=.616.  What  is  the  flow, 
Qt,  in  cubic  feet  per  second? 

Gal  1st.—  By  Rule  25,  or  formula  (81). 

Ratio  of  head  on  bottom   to   that   on   apex,    ra= 


h  __     l       _4 
TT- 


Difference  (f-  iX|)= 


Sum 

Q  =  16.05  X  T.VB-  X  .616  X  ATT  X  W  -  20.  84  cubic 
feet  per  second. 

Cal.  2d.  —  Assuming  that  the  effective  head  is  the 
mean  of  the  given  heads  on  the  apex  and  bottom  of 
the  orifice,  then  will  the  flow  be  the  same  as  found  by 
Cal.  2d,  for  Ex.  35,  viz.,  Q=19.69  cubic  feet  per 
second. 

Cal.  3d.  —  Assume  that  the  true  head  is  on  the  cen- 
ter of  gravity  of  the  opening,  geometrically  con- 
sidered. 

The  center  of  gravity,  as  shown  in  Fig.  14,  is  at 
the  intersection  of  CE  and  DF;  DF  bisecting  BC  in  F. 

CG:CE::2:3;  CG=^; 

AG=AC  +  CG=£'=1  +  fxl.25=1.8334. 

Area  of  opening=5X  1.25-^-2=3.  125. 

Q=.616  X  8.025  X  (1  .8334)2  x  3.  125  =  20.92  cubic 
feet  flow  per  second. 

By  inspection  it  is  seen  that  the  result  by  Cal.  2d 


60 


PKACTICAL    HYDRAULICS. 


is  five  and  one-half  (.055)  of  one  per  cent  too  small, 
and  the  result  by  Gal.  3d  is  nearly  four-tenths  (.0039) 
of  one  per  cent  too  large.  Neither  of  these  empirical 
methods  then  would  satisfy  the  requirements  of  any 
considerable  accuracy. 

FLOW   OF   WATER   THROUGH   A    SUBMERGED   CIRCULAR 
ORIFICE. 

A     A 


FIG.  15. 


Let,  in  Fig.  15,  EDH  represent  a  submerged  circu- 
lar orifice  ;  h—AC,  the  head  on  center  ;  n— angle 
DOE;  r=CD=CE,the  radius;  A=area;  c=coefficient 
of  discharge;  ( 2g) 2=  acceleration  of  gravity;  Q=flow 
in  cubic  feet;  #=any  part  of  r,  estimated  from  C. 
Then,  in  general, 

A=r2— x\  (83) 

Differential  (83),  dA=—2xdx.  (84) 

Head  at  any  point,  B      Ay  B=A — x  cos  n.  (85) 


PRACTICAL    HYDEAULICS.  61 

dQ=  — 2cfi(2^)^i  x  (l—  ~^dx.  (86) 

Integrating  (86)  between  the  limits  x~0,  and  x=v: 


r3  cos  3ra       5rt  cos  4?i 
40/i3  384/1* 

—etc.)  (87) 

Observing  that  the  sum  of  cosines  of  a  complete 
circle  2n  is=0,  and  substituting  the  value  of  (2g)2  = 
8.025,  and  n=3.l416  in  (87): 


(88) 

Making  in  (88),  h=r,  Q=-24.2129c  r*,  (89) 

which  is  identical  with  (57). 


FLOW  OF  WATER  THROUGH  SEMI-CIRCULAR  ORIFICES. 


Let  ii2—  the  mean  of  all  the  cosines  of  the  first  quad- 
rant, and  —  n2=the  mean  of  all  cosines  of  the  second 
quadrant;  then  will  the  mean  of  the  first  and  second 
quadrants  vanish. 

To  determine  the  flow  of  water  through  the  upper 
semi-circle  of  a  submerged  circular  orifice,  substitute 
in  one-half  of  (87),  the  mean  value  of  cos  7i=ii2. 


Q=12.6056C      rl-----etc.     (90) 


62  PKACTICAL  HYDRAULICS. 

To  determine  the  flow  of  water  through  the  lower 
semi-circle  of  a  submerged  circular  orifice,  substitute 
in  one-half  of  (87),  the  mean  value  of  cos  n= — n2. 

Q=12.6056c  AV  (l  + J^  +  ^-^+etc.)  (91 
Ex.  37. — Th?  radius  of  a  circular  opening  is  one 

foot,  the  head  on  the  center  of  the  opening  four  feet, 

and  the  coefficient  of  discharge  .616.     What  is  the 

flow  in  cubic  feet  per  second? 

Gal  1st. — Substituting  the  values  of  r,  h  and  c  in 

formula  (88), 

C=25.2113X  .616X  2  (1- 

105 


(TnrxT*  =— -0019531  \ 

=—.0000191  V 
=_.  0000004  1  =.0019726. 


t 


1_.0019726=.9980274. 
Q=i25.2113X.616X2X.  9980274=31.  00  cubic  feet. 


.  2d.  —  Assume  that  the  true  head  is  on  the  cen- 
ter of  gravity  of  the  opening,  geometrically  con- 
sidered; then  will  the  discharge  be: 

Q=.616X8.025X2X3.14l6=31.06  cubic  feet.— 
Ans. 

Ex.  38.  —  The  radius  of  a  circular  opening  is  one 
foot,  the  head  on  the  center  of  the  opening  four  feet, 
and  the  coefficient  of  discharge  .616.  What  is  the 
flow  in  cubic  feet  per  second  in  the  upper  semi-circle? 


PRACTICAL   HYDRAULICS.  63 

Gal.  —  Substituting  the  values  of  r,  h  and  c  and  ii= 
3.1416  in  formula  (90). 


=—-0530516. 
=—.0019531. 
—  .0001245. 
=—.0000020j  =—.0551312. 

Q'=12.  6056  X.  616  X  2  X.  94487=14.  6739  cubic  feet. 
—  Ans. 

Ex.  39.—  The  radius  of  a  circular  opening  is  one 
foot,  the  head  on  the  center  of  the  opening  four  feet, 
and  the  coefficient  of  discharge  .616.  What  is  the 
flow  in  cubic  feet  per  second  in  the  lower  semi-circle? 

Gal.  —  Substituting  the  values  of  r,  h,  c  and  n= 
3.1416  in  formula  (91  J, 


+  1.  1.0000000 

+  12x321416  =+   .0530516 

—.0019531 


+  ^6o7ki4r6==+   .0001245 

-1024^256  ~.0000020y 

.1.0531761 -.0019551=1.051221. 

Q"=12.6056X. 616X2X1.051221=16.3254     cubic 
feet. — Ans. 


64  PRACTICAL    HYDRAULICS. 

The  value  of  Q"  might  have  been  readily  found  as 
follows: 

Q"=Q—  Q'=31.0—  14.6739=16.326. 

It  will  be  noted  that  formula  (91)  expressed  the  dis- 
charge of  water  through  the  lower  semi-circle  of  a 
submerged  lateral  orifice,  in  which  the  head  is  on  the 
center  ;  whereas  formula  (60)  expresses  the  discharge 
of  water  through  a  semi-circular  weir,  represented  by 
Fig.  10,  in  which  the  head  OA  is  on  the  bottom,  care 
will  need  be  taken  that  these  formulas  be  not  con- 
founded. This  fact  is  rendered  more  apparent  by 
making  in  (91),  h=r,  when  there  results  approxi- 
mately : 


While  with  respect  to  formula  (60), 
Qs=7.693cr*. 


PKACTICAL   HYDRAULICS. 


65 


TABLE  5. 


Square  Boots,  Cubes  of  Square  Roots,  and  Fifth  Powers  of 
Square  Roots  of  Numbers. 


No. 

Square 
Roots. 

Cubes  of 
Square 
Roots. 

5th  Power 
of  Square 
Roots. 

No. 

Square 
Roots. 

Cubes 
of  Square 
Roots. 

5th  Power 
of  Square 
Roots. 

.1 

.316 

.032 

.003 

1. 

1.000 

1.000 

1.000 

.125 

.353 

.044 

.006 

1.25 

1.118 

1.397 

1.747 

.15 

.387 

.058 

.009 

1.50 

1.225 

1.837 

2.756 

.175 

.418 

.073 

.013 

1.75 

1.304 

2.315 

4.051 

.2 

.447 

.089 

.018 

2. 

1.414 

2.822 

5.657 

.225 

.474 

.107 

.024 

2.25 

.500 

3.375 

7.594 

.25 

.500 

.125 

.031 

2.50 

.581 

3.953 

9.882 

.275 

.524 

.144 

.040 

2.75 

.658 

4.560 

12.541 

.3 

.548 

.164 

.049 

3. 

.732 

5.196 

15.590 

.325' 

.570 

.185 

.060 

3.25 

.803 

5.869 

19.041 

.35 

.592 

.207 

.072 

3.50 

.871 

6.547 

22.918 

.375 

.612 

.230 

.086 

3.75 

.936 

7.262 

27.232 

.4 

.633 

.253 

.101 

4. 

2.000 

8.000 

32.000 

.425 

.652 

.277 

.118 

4.25 

2.061 

8.761 

37.236 

.45 

.671 

.302 

.136 

4.5 

2.121 

9.546 

42.957 

.475 

.689 

.327 

.156 

4.75 

2.179 

10.352 

49.174 

.5 

.707 

.353 

.177 

5. 

2.236 

11.180 

55.901 

.525 

.724 

.380 

.200 

5.25 

2.291 

12.029 

63.153 

.55 

.742 

.408 

.224 

5.5 

2.345 

12.898 

70.942 

.575 

.758 

.436 

.251 

5.75 

2.398 

13.783 

79.281 

.6 

.775 

.465 

.279 

6. 

2.449 

14.697 

88.181 

.625 

.791 

.494 

.309 

6.25 

2.500 

15.625 

97.656 

.650 

.806 

.524 

.341 

6.5 

2.550 

16.572 

107.71 

.675 

.821 

.555 

.374 

6.75 

2.598 

17.537 

118.37 

.7 

.837 

.586 

.410 

7. 

2.646 

18.520 

129.64 

.725 

.852 

.617 

.448 

7.5 

2.739 

20.539 

154.04 

.75 

.866 

.650 

.486 

8. 

2.828 

22.627 

181.02 

.775 

.880 

.682 

.529 

8.50 

2.915 

24.781 

210.64 

.8 

.894 

.715 

.572 

9. 

3.000 

27.000 

243.00 

.825 

.908 

.749 

.618 

10. 

3.162 

31.623 

316.23 

.85 

.922 

.784 

.666 

11. 

3.317 

36.483 

401.31 

.875 

.936 

.818 

.717 

12. 

3.464 

41.569 

498.83 

.9 

.949 

.854 

.768 

13. 

3.606 

46.872 

609.33 

.925 

.956 

.890 

.823 

14. 

3.742 

52.383 

733.36 

950 

.975 

.926 

.858 

15. 

3.873 

58.094 

871.41 

.975 

.987 

.963 

.939 

16. 

4. 

64.000 

1024.0 

66 


PBACTICAL   HYDRAULICS. 


TABLE  6. 

Square  Boots  of  Numbers. 


No. 

Square 
Roots. 

No. 

Square 
Roots. 

No. 

Square 
Roots. 

No. 

1  Square 
j    Roots. 

.00 

1.000 

8. 

2.828 

19. 

4.359 

95. 

9.747 

.05 

1.025 

8.1 

2.846 

19.2 

4.382 

96. 

9.798 

.1 

1.049 

8.2 

2.864 

19.4 

4.405 

97. 

9.849 

.15 

1.072 

8.3 

2.881 

19.6 

4.427 

98. 

9.899 

.2 

1.095 

8.4 

2.898 

19.8 

4.45.0 

99. 

9.950 

.25 

1.118 

8.5 

2.915 

20. 

4.472 

100. 

10.000 

.3 

1.140 

8.6 

2.933 

21. 

4.583 

102. 

10.100 

.35 

1.162 

8.7 

2.950 

22. 

4.690 

104. 

10  198 

.4 

1.183 

8.8 

2.966 

23. 

4.796 

106. 

10.295 

.45 

1.204 

8.9 

2.983 

24. 

4.899 

108. 

10.392 

.5 

1.225 

9. 

3. 

25. 

5.000 

110. 

10.488 

.55 

1.245 

9.1 

3.017 

26. 

5.099 

112. 

10.583 

.6 

1.265 

9.2 

3.033 

27. 

5.196 

114. 

10.677 

.65 

1.285 

9.3 

3.050 

28. 

5.292 

116. 

10.770 

.7 

1.304 

-   9.4 

3.066 

29. 

5.385 

118. 

10.863 

.75 

1.323 

9.5 

3.082 

30. 

5.477 

120. 

10.954 

.8 

1.342 

9.6 

3.098 

31. 

5.568 

122. 

11.045 

1.85 

1.360 

9.7 

3.114 

32. 

5.657 

124. 

11.136 

1.9 

1.378 

9.8 

3.130 

33. 

5.745 

126. 

11.225 

1.95 

1.396 

9.9 

3.146 

34. 

5.831 

128. 

11.314 

2. 

1.414 

10. 

3.162 

35. 

5.916 

130. 

11.402 

2.1 

1.449 

10.1 

3.178 

36. 

6.000 

132. 

11.489 

2.2 

1.483 

10.2 

3.194 

37. 

6.083 

134. 

11.576 

2.3 

1.517 

10.3 

3.209 

38. 

6.164 

136. 

11.662 

2.4 

1.549 

10.4 

3.225 

39. 

6.245 

138. 

11.747 

2.5 

1.581 

10.5 

3.240 

40. 

6.325 

140. 

11.832 

2.6 

1.612 

10.6 

3.256 

41. 

6.403 

142. 

11.916 

2.7 

1.643 

10.7 

3.271 

42. 

6481 

144. 

12000 

2.8 

1.673 

10.8 

3.286 

43. 

6.557 

146. 

12.083 

2.9 

1.703 

10.9 

3.302 

44. 

6.333 

148. 

12.166 

3. 

1.732 

11. 

3.317 

45. 

6.708 

150. 

12.247 

3.1 

1  761 

11.1 

3.332 

46. 

6.782 

155. 

12.450 

3.2 

1.789 

11.2 

3.347 

47. 

6856 

160. 

12.649 

3.3 

1.817 

11.3 

3362 

48. 

6.928 

165. 

12.845 

3.4 

1.844 

11.4 

3.376 

49. 

7.000 

170. 

13.038 

3.5 

1.871 

11  5 

3.391 

50. 

7.071 

175. 

13.229 

3.6 

1.897 

11.6 

3  406 

51. 

7.141 

180. 

13417 

3.7 

1.924 

11.7 

3.421 

52. 

7.211 

185. 

13.601 

3.8 

1.949 

11.8 

3.435  , 

53. 

7.280 

190. 

13.784 

3.9 

1.975 

11.9 

3.450 

54. 

7.348 

195. 

13.964 

PKACTICAL    HYDRAULICS. 


67 


TABLE  6.— CONTINUED. 

Square  Roots  of  Numbers. 


No. 

Square 
Roots. 

No. 

Square 
Roots. 

No. 

Square 
Roots. 

I  Square 
Roots. 

No. 

4. 

2.000 

12. 

3.464 

55. 

7.416 

200. 

14.142 

4.1 

2.025 

12.1 

3.479 

56. 

7.483 

205. 

14.318 

4.2 

2.049 

12.2 

3.493 

57. 

7.550 

210. 

14.491 

43 

2.074 

12.3 

3.507 

58. 

7.616 

215. 

14.663 

4.4 

2098 

12.4 

3.521 

59. 

7.681 

220. 

14.832 

4.5 

2.121 

12.5 

3.536 

60. 

7.746 

225. 

15.000 

4.6 

2  145 

12.6 

3.550 

61. 

7.810 

230. 

15.166 

4.7 

2.168 

12.7 

3.564 

62. 

7.874 

235. 

15.330 

4.8 

2.191 

12.8 

3.578 

63. 

7.937 

240. 

15.492 

4.9 

2.214 

12.9 

3.592 

64. 

8.000 

245. 

15.652 

5. 

2.236 

13. 

3.606 

65. 

8.062 

250. 

15.811 

5.1 

2.258 

13.2 

3.633 

66. 

8.124 

260. 

16.125 

5.2 

2.280 

13.4 

3.661 

67. 

8.185 

270. 

16.432 

5.3 

2.302 

13.6 

3.688 

68. 

8.246 

280. 

16.733 

5.4 

2.324 

13.8 

3.715 

69. 

8.307 

290. 

17.029 

5.5 

2.345 

1    14. 

3.742 

70. 

8.367 

300. 

17.320 

5.6 

2.366 

14.2 

3.768 

71. 

8.426 

310. 

17.607 

5.7 

2.387 

14,4 

3.795 

72. 

8.485 

320. 

17.889 

5.8 

2.408 

14.6 

3.821 

73. 

8.544 

330. 

18.166 

5.9 

2.429 

14.8 

3.847 

74. 

8.602 

340. 

18.439 

6. 

2.449 

15. 

3.873 

75. 

8.660 

350. 

18.708 

6.1 

2.470 

15.2 

3.899 

76. 

8.718 

360. 

18.974 

6.2 

2.490 

15.4 

3.924 

77. 

8.775 

370. 

19.235 

6.3 

2.510 

1    15.6 

3.950 

78. 

8.832 

380. 

19.494 

6.4 

2.530 

15.8 

3.975 

79. 

8.888 

390. 

19.748 

6.5 

2.550 

16. 

4.000 

80. 

8.944 

400. 

20.000 

6.6 

2.569 

16.2 

4.025 

81. 

9.000 

410. 

20.248 

6.7 

2.588 

16.4 

4,050 

82. 

9.055 

420. 

20.494 

6.8 

2.608 

16.6 

4.074 

83. 

9.110 

430. 

20.736 

6.9 

2.627 

16.8 

4.099 

84. 

9.165 

440. 

20.976 

7. 

2.646 

17. 

4.123 

85. 

9.220 

450. 

21.213 

7.1 

2.665 

17.2 

4.147 

86. 

9.274 

460. 

21.448 

7.2 

2.683 

17.4 

4.171 

87. 

9.327 

470. 

21.679 

7.3 

2.702 

17.6 

4.195 

88. 

9.380 

480. 

21.909 

7.4 

2.720 

17.8 

4.219 

89. 

9.434 

490. 

22.136 

7.5 

2.739 

18. 

4.243 

90. 

9.487 

500. 

22.361 

7.6 

2.757 

18.2 

4266 

91. 

9.539 

525. 

22.913 

7.7 

2.775 

18.4 

4.290 

92. 

9.592 

550. 

23.452 

7.8 

2.793 

18.6 

4.313 

93. 

9.644 

575. 

23.979 

7.9 

2.811        18.8 

4.336 

94. 

9.695 

600. 

24.495 

68  PRACTICAL    HYDRAULICS. 

FLOW  OF  WATER  THROUGH  VERTICAL,  RECTANGULAR 
ORIFICES  IN  THIN  PARTITIONS. 

The  assumption  that  the  mean  velocity  of  a  stream 
of  water  flowing  through  a  vertical  rectangular  ori- 
fice is  at  the  middle  of  the  opening,  has  been  shown 
by  equation  (22)  to  be  not  strictly  true.  But,  owing 
to  its  simplicity  of  application,  and  its  close  approxi- 
mation to  the  truth,  hydraulicians,  for  the  most  part, 
are  wont  to  adopt  it,  and  to  correct  the  error  involved 
by  coefficients  obtained  by  experiment. 

Table  7,  derived  from  Fanning's  Hydraulic  En- 
gineering, embraces  a  wTide  range  of  coefficients  so 
determined.  Thus,  it  is  suited  to  heads  of  water 
from  two-tenths  (.2)  of  a  foot  to  fifty  (50)  feet,  and 
to  orifices  one  foot  wide,  whose  hights  vary  from  four 
(4)  feet  to  one  and  one-half  (1J)  inches. 


PRACTICAL    HYDRAULICS. 


69 


TABLE  7. 


Flow  of  Water  per  second  through  rectangular  orifices  in  thin 

vertical  partitions,  and  the  coefficients  employed 

in  the  computation. 


Head 
on 
Center. 

Coeffi- 
cient. 

4  feet 
high;   1 
ft.  wide 

Coeffi- 
cient. 

2  feet 
high;   1 
ft.  wide. 

Coeffi- 
cient. 

1£  feet 
high  ;   1 
ft.  wide 

Coeffi- 
cient. 

Ifoot 
high;  1 
ft.  wide 

0.6 

.598 

3.72 

0.7 

.599 

4.02 

0.8 

.613 

6.60 

.600 

4.31 

0.9 

.613 

7.01 

.601 

4.57 

1.0 

.614 

7  39 

.601 

4.87 

1.25 

.619 

11.11 

.614 

8.26 

.602 

5.29 

1.50 

.619 

12.16 

.614 

9.06 

.603 

5.92 

1.75 

.619 

13.13 

.615 

9.79 

.603 

6.40 

2.00 

.618 

14,04 

.614 

10.45 

.604 

6.85 

2.25 

.618 

14.89 

.614 

10.96 

.604 

7.27 

2.50 

.629 

31.92 

.618 

15.67 

.614 

11.66 

.604 

7.67 

2.75 

.628 

33.43 

.617 

16.43 

.614 

12.24 

.605 

8.05 

3.00 

.627 

34.75 

.617 

17.15 

.613 

12.78 

.605 

8.41 

3.50 

.625 

37.54 

.616 

18.49 

.612 

13.79 

.605 

9.08 

4.00 

.625 

40.09 

.615 

19.74 

.611 

14.71 

.605 

9.97 

4.50 

.623 

42.39 

.614 

20.90 

.610 

15.58 

.604 

10.29 

5.00 

.621 

44.55 

.612 

21.98 

.609 

16.48 

.604 

10.84 

6.00 

.616 

48.42 

.609 

23.96 

.606 

17.88 

.602 

11.84 

7.00 

.612 

52.23 

.606 

25.75 

.604 

19.23 

.601 

12.76 

8.00 

.609 

55.29 

.604 

27.39 

.602 

20.50 

.601 

13.64 

9.00 

.606 

58.35 

.602 

28.98 

.601 

21.72 

.601 

14.47 

10.00 

.604 

61.26 

.602 

30.53 

.601 

22.88 

.601 

15.25 

15.00 

.604 

75.09 

.602 

37.42 

.601 

28.02 

.601 

18.68 

20.00 

.605 

86.78 

.602 

43.24 

.601 

32.37 

.601 

21.50 

25.00 

.605 

99.06 

.603 

48.39 

.601 

36.19 

.601 

24.12 

30.00 

.605 

106.46 

.603 

53.34 

.602 

39.68 

.601 

26.43 

35.00 

.606 

115.08 

.694 

57.35 

.602 

4288 

.601 

28.55 

40.00 

.607 

123.13 

.605 

61.36 

.603 

45.86 

.602 

30.53 

45.00 

.606 

130.39 

.605 

65.14 

.603 

48.68 

.602 

32.39 

50.00 

.609 

138.12 

.606 

67.21 

.603 

51.36 

.602 

34.15 

Mean. 

Mean. 

Mean. 

Mean. 

.614 



.610 



.608 



.602 



70 


PRACTICAL    HYDRAULICS. 


TABLE  7. 


Plow  of  Water  per  second  through  rectangular  orifices  In  thin 

vertical  partitions,  and  the  coefficients  employed 

in  the  computation. 


Head 
on 
Center. 

Coeffi- 
cient. 

9  fett 
high;    1 
t.  wide, 
cu.  ft. 

Coeffi- 
cient. 

6  feet 
high;    1 
ft.  wide, 
cu.  ft. 

Coeffi- 
cient. 

3  feet 
high;    i 
ft.  v  ide 
cu.  ft. 

foeffi- 
c  ent. 

li  feet 
high;    1 
ft.  wide, 
cu.  ft. 

0.2 

.633 

.28 

0.3 

.629 

.69 

.633 

.35 

0.4 

.614 

1.56 

.631 

.80 

.633 

.40 

0.5 

.605 

2.57 

.615 

1.74 

.631 

.89 

.633 

.45 

0.6 

.606 

2.83 

.616 

1.91 

.632 

.98 

.633 

.49 

0.7 

.607 

3.06 

.616 

207 

.632 

.06 

.633 

.53 

0.8 

.608 

3.27 

.617 

2.21 

.632 

.14 

.633 

.59 

0.9 

.609 

3.48 

.617 

2.35 

.632 

.20 

.632 

.60 

1.0 

.609 

3.67 

.617 

2.48 

.632 

.26 

.632 

.63 

1.25 

.610 

4.02 

.617 

2.71 

.632 

.39 

.631 

.69 

1.50 

.610 

4.50 

.617 

3.03 

.631 

1.55 

.630 

.77 

1.75 

.610 

4.86 

.617 

3.27 

.631 

1.67 

.630 

.83 

2.00 

.610 

5.20 

.617 

3.50 

.630 

1.79 

.629 

.89 

2.25 

.610 

5.51 

.616 

3.71 

.629 

1.89 

.629 

.95 

2.50 

.610 

5.81 

.616 

391 

.628 

1.99 

.628 

1.00 

2.75 

.610 

6.09 

.616 

4.10 

.627 

2.09 

.627 

1.04 

3.00 

.610 

6.36 

.615 

4.27 

.627 

2.18 

.627 

1.09 

3.5 

.609 

6.36 

.615 

461 

.625 

2.35 

.625 

1.17 

4.00 

.609 

7.32 

.614 

4.92 

.624 

2.50 

.624 

1.25 

4.5 

.607 

7.75 

.613 

5.21 

.622 

2.65 

.622 

1.32 

5.00 

.606 

8.16 

.611 

549 

.620 

2.78 

.620 

1.39 

6.00 

.604 

8.91 

.609 

5.98 

.615 

3.03 

.615 

1.51 

7.00 

.603 

9.61 

.606 

6.43 

.611 

3.24 

.611 

1.62 

8.00 

.602 

10.25 

.603 

6.84 

.607 

3.45 

.609 

1.71 

9.00 

.602 

10.86 

.602 

7.25 

.605 

3.64 

.607 

1.83 

10.00 

.601 

11.44 

.601 

7.62 

.603 

3.83 

.606 

1.92 

15.00 

.601 

14.01 

.601 

9.34 

.603 

4.69 

.607 

2.36 

20.00 

.601 

16.18 

.602 

10.80 

.604 

5.42 

.607 

2.72 

25.00 

.602 

18.10 

.602 

12.08 

.604 

6.06 

.608 

3.05 

30.00 

.602 

19.84 

.603 

13.47 

.604 

6.64 

.609 

3.35 

35.00 

.602 

21.44 

.603 

14.31 

.605 

7.18 

.610 

3.62 

40.00 

.603 

22.94 

.604 

15.32 

.606 

7.68 

.611 

3.79 

45.00 

.603 

24.35 

.604 

16.26 

.606 

8.16 

.613 

4.12 

50.00 

.604 

25.68 

.605 

17.16 

.607 

8.61 

.614 

4.35 

Mean. 

Mean. 

Mean. 

Mean. 

.606 

.611 



.620 



.622 

PEACTICAL   HYDRAULICS. 


An  inspection  of  Table  7  discloses  that  the  coeffi- 
cient of  flow  is  variable,  both  with  respect  to  the 
head  of  water  and  form  of  orifice. 

Thus,  the  orifice  being  '  'four  feet  high,"  the  maxi- 
mum coefficient  .629,  is  due  a  head  of  2.50  feet ; 
thence  the  coefficient  gradually  diminishes  to  .  604,  as 
the  head  increases  to  10  feet;  thence  is  constant  to  15 
feet:  thence  gradually  increases  to  .609,  with  the  in- 
crease of  the  head  to  50  feet. 

In  the  other  given  orifices,  variations  obtain,  but  to 
a  less  extent. 

With  respect  to  the  variation  of  coefficients  arising 
from  the  form  of  orifice,  it  will  be  seen,  by  running 
the  eye  horizontally  to  the  right,  from  and  for  any 
given  head,  that  the  values  of  the  coefficients  diminish 
as  the  hights  of  the  orifices  decrease  from  four  feet  to 
one  foot,  and  increase  as  the  hights  of  the  orifice  de- 
crease from  one  foot  to  one  and  one-half  (1J)  inches. 
In  illustration  take  several  heads,  as  3,  10,  25,  50  feet, 
and  the  coefficients  due  the  several  forms  of  orifice. 


Head. 
Feet. 

4'xl' 
Coef. 

2'xl' 
Coef. 

li'  x  1' 
Coef. 

1'xl' 
Coef. 

9"  x  1' 
Coef. 

6x1' 
Coef. 

3"  x  1' 
Coef. 

H'xl" 
Coef. 

3 
10 
25 
50 

.627 
.604 
.605 
.6°9 

.617 
.602 
.603 
.fi°6 

.613 
.601 
.601 
.603 

.605 
.601 
.601 
.602 

.610 
.601 
.602 
.604 

.615 
.601 
.602 
.605 

.627 
.603 
.604 
.607 

.627 
.606 
.608 
.614 

72  PRACTICAL    HYDRAULICS. 

TO  FIND  THE  FLOW  OF  WATER  IN  CUBIC  FEET  PER 
SECOND  THROUGH  VERTICAL  RECTANGULAR  ORIFICES  IN 
THIN  VERTICAL  PARTITIONS  BY  TABLE  7,  THE  HEAE  ON 
CENTER  AND  SIZE  OF  OPENING  BEING  MADE. 

Rule  26.  —  In  "head  on  center"  column,  Table  7, 
find  the  given  head,  opposite  which,  in  column  headed 
by  the  given  hight  of  orifice,  will  be  found  the  flow 
for  an  orifice  one  foot  wide,  which  multiply  b}^  the 
given  width  in  feet. 

Ex.  40.  —  The  head  being  ten  feet,  and  orifice  four 
feet  wide  and  nine  inches  high,  what  is  the  flow  per 
second? 

Cal.  —  In  column  "9"  high  1  foot  wide,"  opposite  10 
feet  in  "head  on  center"  column,  will  be  found  11.44 
cubic  feet,  which,  multiplied  by  four  feet,  the  given 
width,  gives: 

11.44X4=45.76  cubic  feet.—  Ans. 

The  heads,  sizes  of  orifices,  and  the  computed  flow 
of  water  given  in  Table  7,  will  be  found  highly  con- 
venient for  ready  reference  in  a  great  number  of  cases, 
but  are  seen  to  be  too  limited  to  fully  meet  the  re- 
quirements of  practice.  Indeed,  a  table  sufficiently 
ample  for  that  purpose  would  be  too  un  wieldly  for  use. 

The  general  formula  for  the  low  of  water  per  sec- 
ond through  vertical  rectangular  orifices  in  thin  par- 
titions, is: 

(92) 


In  which  Q  denotes  the  flow  in  cubic  feet  ;  c,  coefficient 


PKACTICAL   HYDRAULICS.  73 

of  discharge;  a,  the  area  of  the  orifice  in  square  feet; 
and  ti,  the  head  on  the  center  of  the  orifice:  h!  is  equal 
to  the  half  sum  of  the  respective  heads  on  the  bottom 
and  top  of  the  orifice,  as  seen  in  equation  (21). 

In  case  the  hight  of  the  orifice  and  the  head  on  its 
top  are  given,  then  hr  is  equal  to  the  sum  of  the  given 
head  and  half  the  hight  of  the  opening;  or  if  the  hight 
of  the  opening  and  the  head  on  its  bottom  are  given, 
then  hr  is  equal  to  the  difference  between  the  given  head 
and  half  the  hight  of  the  orifice. 

TO  FIND  THE  FLOW  OF  WATER  IN  CUBIC  FEET  PER 
SECOND,  THROUGH  VERTICAL  RECTANGULAR  ORIFICES  IN 

THIN  PARTITIONS. 

* 

Rule  27. — Multiply  8.025  times  the  square  root  of 
the  head  on  the  center  of  the  orifice,  by  the  product  of 
the  area  of  the  orifice  and  the  coefficient  of  discharge. 

Rule  27  corresponds  to  formula  (92). 

With  respect  to  the  "square  root  of  the  head,"  and 
"the  coefficient  of  discharge,"  contemplated  in  Rule 
27,  it  will  be  remembered  that  Table  6  gives  the  square 
roots  of  numbers  likely  to  be  required,  and  Table  7, 
the  coefficients  of  discharge.  In  finding  a  proper  co- 
efficient of  discharge,  in  case  the  given  hight  of  orifice 
is  found  in  Table  7,  the  coefficient  corresponding  to 
that  hight  and  to  the  given  head  is  to  be  employed; 
but  in  case  the  given  hight  of  orifice  is  an  inter- 
mediate, or  lies  between  the  hights  contained  in  the 


74  PRACTICAL    HYDRAULICS. 

table,  its  coefficient  will  need  be  computed.  The 
tabulated  coefficients  are,  in  fact,  ordinates  of  curves, 
determined  by  experiment. 

In  determining  these  intermediate  ordinates  or  co- 
efficients between  any  two  adjacent  hights  in  Table  7, 
as  4  feet  and  2  feet,  1.5  feet  and  1  foot,  no  appreciable 
error  will  occur  by  substituting  a  right  line  for  a 
curve.  The  determination  of  the  intermediate  coeffi- 
cients will  then  be  effected  by  arithmetical  differences. 
In  illustration:  let  it  be  required  to  find  the  coefficient 
due  a  head  of  2.5  feet,  and  orifice  3.5  feet  high. 

Now  3.5  is  between  the  adjacent  hights,  4  and  2 
feet,  in  Table  7.  The  respective  coefficients  due  a 
head  of  2.5  feet  are  .629  in  "4  feet  high"  column,  and 
.618  in  "2  feet  high"  column. 

Difference  of  hights,  4—2=2  feet. 

Difference  of  greater  and  given  hights,  4 — 3.5 —.5  feet. 
Quotient  of  these  differences,  2-v-.5— 4  divisor. 

Difference  of  coefficients,  .629— .618=. Oil 

Arithmetical  difference  sought,  .Oll-j-4=.003  nearly. 
Coefficient  due  3.5  feet,  .629— .003=.626 

The   intermediate    coefficients   corresponding   to    3 
feet  and  2.5  feet  are  now  readily  found.     Thus: 
Coefficient  due  3  feet,  .626— .003=.623 

Coefficient  due  2.5  feet,  .623— ;003=.620 

EXAMPLES  AND  CALCULATIONS. 
Ex.  41.— An  orifice  is  3,5  feet  wide,  1.25  feet  high 


PRACTICAL   HYDRAULICS.  75 

and  the  head  on  its  center  is  7  feet.    What  is  the  flow 
in  cubic  feet  per  second? 

Cal— By  Table  6,  square  root  of  7  feet=2.646. 

By  Table  7,  coefficient  due  7  feet;  for  orifice,  1.5 
feet  high=.604;   for  orifice,  1  foot  high^  .601. 
Difference  of  tabulated  hights,  1 . 5 — 1 = .  5 

Difference  of  greater  and  given  hights,  1.5 — 1.25=.  25 
Quotient  of  these  differences,  .  5  -=-  .25 — 2  div . 

Difference  of  coefficients,  .  604—.  60l=.003 

Arithmetical  difference,  .003-^-2=.0015 

Coefficient  due  1.25  feet,  .604— .0015=. 6025 

Area  of  orifice',  3.5X1.25=4.375  square  feet. 

Flow=8. 025  X .  6025  X4. 375  X  2. 646=55. 97  cubic 
feet. — Ans. 

Ex.  42. — Given  the  head  on  the  bottom  of  a  rec- 
tangular orifice  12  feet,  the  head  on  its  top  11  feet, 
and  the  width  of  orifice  4  feet,  what  is  the  flow  in 
cubic  feet  per  second? 

Cal.— Head  on  center =^=11. 5  feet. 

By  Table  6,  square  root  of  head  on  center =3. 391 
feet. 

Hight  of  orifice=12— 11=1  foot. 

By  Table  7,  coefficient  due  head  of  11.5  feet,  and 
orifice  1  foot  high =.601. 

It  will  be  observed  that  the  coefficient  is  constant 
for  head  from  10  to  1 5  feet,  inclusive. 

Area  of  orifice=4Xl=4  square  feet. 

Flow=8.025X.60lx4x3.391=65.42  cubic  feet.— 
Ans. 


76  PRACTICAL   HYDRAULICS. 

Ex.  43. — The  head  on  the  top  of  a  rectangular  ori- 
fice 6  inches  high  and  6  feet  wide  being  7.25  feet, 
what  is  the  flow  in  cubic  feet  per  second? 

CaL— Half  the  hight  of  orifice  6"-*-2=  3"— .25  feet, 

Head  on  center =7. 25  +  . 25=7.5  feet. 

By  Table  7,  coeflicient  due  head  of  7  feet=.606. 

Coefficient  due  head  of  8  feet=.603. 

Mean  coefficient  on  that  due  7.5  feet=.6045. 

By  Table  6,  square  root,  7.5  feet— 2.739. 

Area  of  orifice,  6X  .5=3  square  feet. 

Flow=8. 025  X  .6045  X  3  X  2.739=29.89  cubic  feet.— 
Ans. 

Ex.  44. — The  head  on  the  bottom  of  a  rectangular 
orifice  9  inches  high  and  3  feet  wide,  being  15.875 
feet,  what  is  the  flow  in  cubic  feet  per  second  ? 

CM.— Half  the  hight  of  orifice,  9"-^2=4.5"=.375 
feet. 

Head  on  center  =15. 97 5— .375=15.6  feet. 

By  Table  7,  coefiicient  due  head  of  15.875  feet,  and 
orifice  9"  high=.601. 

Observe  that  the  coefficient  is  constant  from  10  feet 
to  20  feet,  inclusive. 

By  Table  6,  square  root  of  15.6=3.95  feet. 

Area  of  orifice=3X.75=2.25  square  feet. 

Flow=8.025X. 601X2.25X3. 95=42.86  cubic  feet. 
— Ans. 

The  preferable  unit  for  measuring  the  flow  of  water 
is  1  cubic  foot,  but  so  widely  is  the  "miner's  inch," 
employed  in  California  as  a  unit  of  measure,  that  we 
cannot  well  pass  it  in  silence. 


PRACTICAL   HYDRAULICS.  77 


"MINER'S  INCH." 


The  term  "miner's  inch"  is  employed  to  express  that 
quantity  of  water  which,  under  a  given  head  or  pres- 
sure, as  4,  7,  9,  etc.,  inches,  will  flow  through  each 
square  inch  of  a  discharge  opening ;  or,  in  other  words, 
which  will  flow  through  each  square  inch  of  cross  sec- 
tion of  a  stream  of  water. 

The  quantity  of  water  so  flowing  in  a  minute,  an 
hour,  24  hours,  etc.,  is  designated  minute  inch,  hour 
inch,  24:-hour  inch,  etc.,  according  to  the  length  of 
time  specified. 


STATUTORY  MINER'S  INCH. 


Under  the  head,  "Water  Rights,"  the  Civil  Code  of 
the  State  of  California,  Sec.  1415,  provides  in  these 
words,  "That  he  (the  locator)  claims  the  water  there 
flowing  to  the  extent  of  (giving  the  number)  inches, 
measured  under  a  four-inch  pressure." 

On  this  data,  the  value  of  the  statutory  miner's 
inch,  the  mean  coefficient  of  discharge  being  in  prac- 
tice .6216,  is  as  follows: 

For  one  second  (second  inch),  0.02  cubic  feet. 

For  one  minute  (minute  inch),  1.20  cubic  feet. 


78  PRACTICAL    HYDRAULICS. 

For  one  hour  (hour  inch),  72.00  cubic  feet. 

For  24  hours  (24-hour  inch),  1728  cubic  feet. 

If  a  cubic  foot  be  divided  by  the  flow  in  one  sec- 
ond, there  will  result  the  number  of  miner's  inches 
whose  discharge  is  equal  to  a  cubic  foot  per  second. 
Thus,  1^-. 02=50  statutory  miner's  inches;  that  is, 
fifty  statutory  miner's  inches  are  equal  to  one  cubic 
foot  flow  per  second. 


NORTH  BLOOMFIELD,  ETC.,  MINER'S  INCHES. 


At  the  North  Bloomfield,  Milton  and  Columbia  Hill 
hydraulic  mines,  the  water  is  measured  in  its  flow 
through  a  rectangular  orifice  50  inches  long,  2  inches 
wide,  and  under  a  pressure  of  7  inches  on  the  center  of 
the  opening.  The  flow  per  square  inch  of  orifice,  for 
24  hours,  due  this  head,  as  given  me  by  Hamilton 
Smith,  Jr.,  C.  E.,  formerly  chief  engineer  of  the 
North  Bloomfield  Works,  and  president  of  the  Miners' 
Association,  is  2230  cubic  feet;  of  which  the  coefficient 
of  discharge  is  found  to  be  .6064.  Mr.  Smith's  ex- 
periments made  with  a  module  of  equal  dimensions 
under  a  7-inch  head,  at  Columbia  Hill  in  1874,  found, 
as  stated  by  Aug.  J.  Bowie,  Jr.,  M.  E.,  in  an  article 
entitled  "Bowie  on  the  Measurement  and  Flow  of 
Water,"  found  the  value  of  the  24-hour  inch  to  be 
2260.8  cubic  feet,  and  the  coefficient  of  discharge  to  be 
.616.  Mr.  Bowie,  in  the  article  referred  to,  gives,  in 


PBACTICAL    HYDRAULICS.  79 

addition,  substantially  the  following  data,  with  respect 
to  the 

SMARTSVILLE  MINER'S  INCH. 


Hight  of  orifice,  4  inches;  head  on  center,  9  inches; 
value  of  24-hour  inch,  2534.4  cubic  feet;  coefficient  of 
discharge,  .6078. 


SOUTH  YUBA  CANAL  INCH. 


Hight  of  orifice,  2  inches;  head  on  center,  6  inches. 

And  with  respect  to  a  series  "of  experiments  made 
by  himself  at  La  Grange  with  an  orifice  12  inches 
high,  12.75  inches  wide,  under  a  pressure  of  6  inches 
on  the  top  of  the  orifice,  or  head  of  one  foot  on  the 
center.  The  mean  of  which  experiments  gave  as  the 
value  of  otfe  miner's  inch  for  24  hours,  2159.146  cubic 
feet;  effective  coefficient  of  efflux,  .5905.  The  flow 
through  this  module  was  assumed  equal  to  200  miner's 
inches. 

A  comparison  shows  that  these  coefficients  of  dis- 
charge approximate  closely  those  given  in  Table  7, 
obtained  on  equal  data.  The  results  of  these  experi- 
ments also  clearly  show  that  the  value  of  a  coefficient 
of  discharge  depe.nds,  among  other  things,  upon  the 
form  of  the  module.  In  illustration,  the  module  be- 


80 


PRACTICAL    HYDRAULICS. 


ing  50  inches  long,  2  inches  wide,  the  coefficient  of 
discharge  is  found  to  be  .5905.  Should  we  estimate 
the  effect  of  the  difference  between  the  given  heads 
(7  inches  and  1  foot)  on  the  coefficients  of  discharge, 
there  would  result  .5885  instead  of  .5905. 

The  variety  of  values  comprised  in  the  term, 
miner's  inch,  as  employed  in  California,  is  often  a 
source  of  no  little  annoyance  and  confusion.  To  aid 
in  overcoming  this  difficulty,  Table  8  prepared  from 
Table  7,  is  given.  Each  result  so  obtained  is  a  mean 
of  the  experiments  of  the  world's  ablest  hydraulicians. 

TABLE  8. 


Flow  of  water  through  rectangular  orifices  due  Miner's 
Inches  of  different  values. 


Head 

Orifice. 

|    ec.  inch 

Min.  inch 

Hour  inch 

24-hour 

1  Cub.  Ft. 

on 
Cent. 

High 

Wide. 

Coef.  j     Flow. 

F.ow. 

Flow. 

inch  Flow. 

Flow 
j;er  Sec. 

In. 

In. 

In. 

Cub.  Feet. 

Cub.  Feet. 

CuH.  Feet. 

Cub.  Pfeet. 

Miner'sln. 

3 

2 

12. 

.631 

.01758 

.055 

63.29 

1519. 

58.87 

4 

2 

12. 

.631 

.02030 

.218 

73.10 

1754. 

49.24 

5 

2 

12. 

.632 

.02274 

.364 

81.85 

1964. 

43.98 

6 

2 

12. 

.632 

.02490 

.494 

89.65 

2152. 

40.15 

7 

2 

12. 

.632 

.02690 

.614 

96.84 

2324. 

37.17 

8 

2 

12. 

.632 

.02876 

.725 

103.53 

2484. 

34.77 

9 

4 

12. 

.624 

.03011 

1.807 

108.42 

2602. 

33.20 

10 

6 

12. 

.617 

.03139 

1.883 

113.00 

2711. 

31.85 

11 

9 

12. 

.609 

.03249 

1.949 

116.98 

2807. 

30.77 

*12 

12 

12.75 

.601 

.02562 

1.537 

92.24 

2214. 

39.03 

*The  flow  due  the  given  opening,  12"  x  12  75"  =  153  square  inches, 
divided  by  200,  has  been  proposed,  as  hereinbefore  stated,  as  the 
standard  miner's  inch.  Its  adoption  seems  to  be  but  local. 


PRACTICAL  HYDRAULICS.  81 


EXAMPLES  AND  CALCULATIONS. 


Ex.  45. — A  water  right  location  is  made  for  6000 
miner's  inches.  What  is  the  equivalent  flow  in  cubic 
feet  per  second? 

Gal.  1st. — The  statutory  miner's  inch  is  estimated, 
as  stated,  under  a  4-inch  pressure. 

By  Table  8,  opposite  4  inches  in  "head  on  center" 
column,  find  49.24  miner's  inches  in  "1  cubic  foot  flow 
per  second"  column,  6000^-49.24=121.6  cubic  feet. 
— Ans. 

CaL  2d. — For  the  most  part  in  practice,  50  miner's 
inches  measured  under  a  4-inch  pressure,  are  adopted 
as 'equal  to  a  flow  of  one  cubic  foot  of  water  per  sec- 
ond. This  results,  as  shown  in  discussing  the  statu- 
tory miner's  inch,  from  taking  the  mean  coefficient 
.6216  for  different  heads,  instead  of  the  tabulated  co- 
efficient .632  for  the  given  4-inch  head. 

Whence,  6000-^50=120  cubic  feet.— Ana. 

Ex.  46. — In  a  water  right  claim  of  5000  miner's 
inches,  measured  under  a  4-inch  pressure,  are  how 
many  North  Bloomfield  miner's  inches — miner's  inches 
measured  under  a  7- inch  head? 

Col.  1st. — By  Table  8,  the  value  of  a  second  inch, 
under  a  4-inch  head,  is  .0203  cubic  feet  flow;  and  the 
value  of  a  second  inch,  under  a  7-inch  head,  is  .0269 
cubic  feet  flow. 


82  PRACTICAL   HYDRAULICS. 

Whence,  .0203X5000-^.0269=3773  miner's  inches 
— Ans. 

Cal.  2d. — In  the  discussion  of  the  miner's  inch,  it 
has  been  shown  that  in  common  practice  the  value  of 
the  24-hour  inch,  under  a  4-inch  head,  is  1728  cubic 
feet;  and  under  a  7-inch  head  at  North  Bloomneld  is 
2230  cubic  feet. 

Whence,  1728x5000-+- 2230=- 3874  miner's  inches. 
— Ans. 

Ex.  47. — In  2000  miner's  inches,  through  a  rectan- 
gular opening  2  inches  high,  and  under  a  6-inch  pres- 
sure, as  employed  at  the  South  Yuba  canal,  are  how 
many  miner's  inches  flowing  through  a  rectangular 
opening  4  inches  high  and  under  a  9-inch  pressure,  as 
adopted  at  Smartsville? 

Cal.  1st. — As  the  result  will  be  the  same,  whether 
the  calculation  be  made  in  second,  minute,  hour,  or 
24-hour  miner's  inches,  let  the  24-hour  inch  be  em- 
ployed ;  then  by  Table  8,  under  a  6 -inch  pressure 
through  an  opening  2  inches  high,  the  value  is  2152 
cubic  feet ;  and  under  a  9-inch  pressure  through  an  open- 
ing 4  inches  high,  the  value  is  2602  cubic  feet;  whence, 
2000x2152-^-2602=1654  Smartsville  miner's  inches. 

Cal.  2d. — Under  the  heading  Smartsville,  Bowie 
"On  Measurement  and  Flow  of  Water,"  makes  that 
miner's  inch,  2534.4,  due  coefficient  of  discharge  .6078. 
instead  of  .624,  as  adopted  in  Table  8. 

Whence,  2000X2152  +  2534.4=1698  Smartsville 
miner's  inches. — Ans. 

Cal.  3d. — Table  8  shows  that  the  coefficient  due  a 


PRACTICAL    HYDRAULICS.  83 

6  -inch  head,  and  opening  2  inches  high,  is  .632,  and 
that  the  coefficient  due  a  9-inch  head,  and  opening  4 
inches  high,  is  .624.  Now,  as  commonly  practiced, 
the  "mean"  coefficient  .62  would  be  employed;  so  that, 
the  result  sought  would  depend  upon  the  square  root 
of  the  ratio  of  the  given  heads,  9  inches=.75  feet,  and 
6  inches-=.5  feet;  thus,  by  Table  5, 


.8165X2000=1633  Smartsville  miner's  inches.- 
Ans. 

EXAMPLES  AND  CALCULATIONS. 

Ex.  48.  —  The  head  being  2.25  feet  on  the  center  of 
a  circular  orifice  .0328  feet  diameter,  what  is  the  dis- 
charge in  cubic  feet  per  second? 

Gal.  1st.  —  Rule  27  is  equally  applicable  to  rectangu- 
lar and  circular  orifices. 

By  Table  6  the  square  root  of  given  head  2.25  feet— 
1.5  feet. 

Area  of  given  orifice  .0323  feet  diameter  is  equal  to 
the  square  of  the  diameter,  multiplied  by  .7854  ; 
(.0328)2X.  7854=  .000845  square  feet. 

By  Table  9,  coefficient  of  discharge  due  a  head  of 
2.25  feet,  according  to  Castel,  is  approximately  equal 
to  .673  ;  then  by  Rule  27, 

.873X8.025Xl.5X.  000845=.  00685  cubic  feet.— 
Ans. 

Cal.  2d.  —  According  to  Weisbach,the  coefficient  due 
a  head  of  2.25  feet,  and  orifice  .0328  feet  diameter,  is 


84  PRACTICAL  HYDRAULICS . 

approximately  .628.     Employing  this  coefficient  in- 
stead of  .673, 

.628X 8.025X1. 5X- 000845=. 00639    cubic   feet.— 
Ans. 

•  Weisbach  observes,  that  "for  square  orifices  from  1 
to  9  square  inches  area,  wrth  from  7  to  21  feet  head  of 
water,  according  to  the  experiments  of  Bossut  and 
Michelotti,  the  mean  coefficient  of  efflux  is  ?7i=.610; 
for  circular  ones  of  from  J  to  6  inches  diameter,  with 
from  4  to  21  feet  head  of  water,  m=.6l5." 

A  mean  of  the  coefficients  of  Table  9  is  equal  to  .62 
nearly. 

In  ordinary  practice  this  is  employed.  When 
greater  accuracy  is  required,  reference  will  need  be  had 
to  Table  9. 

PARTIAL  CONTRACTION. 

Experiments  show  that  if  contraction  be  suppressed, 
the  flow  of  water  through  an  orifice  will  be  increased 
accordingly. 

Let  7i-— the  ratio  between  the  entire  perimeter  of  an 
orifice  and  the  part  suppressed — that  is,  if  p  denote 
the  entire  perimeter,  and  pt  the  part  suppressed,  then 

*=?• 

c=coefficient  of  discharge  due  perfect  contraction. 

cn— coefficient  of  discharge  due  partial  contraction. 

#= a  number  deduced  from  experiment,  which,  be- 
ing multiplied  by  the  product  of  the  ratio,  n,  and  the 
coefficient,  c,  gives  cxn,  the  increase  due  partial  con- 
traction; thus,  Cn==c  (1+ajw). 


PRACTICAL  HYDRAULICS. 


85 


TABLE  9. 

Coefficients  for  the  Flow  of  Water  through  circular  orifices. 
Extracts  from  D'Aubiussoc,  Fannies  and  Weisbach. 


OBSERVERS. 


Diam      1      Heads. 
Feet.        '     Feet. 


Mariotti 0.0223  5.8712 

.0223  25.9120 

Castel !     .0328  2.1320 

«      J     .0328  1.0168 

0492  0.4526 

.0492  0.9840 

Eytelwine .* 0856  2.3714 

Bossut 0889  42640 

Michelo^ti |     .0889  7  3144 

Castel i     .0984  05510 

Veatari '....!     .1345  2.8864 

Bossut I     .1771  12.4968 

Michelotti 1771  7.2160 

.2657  73472 

.2657  12.4968 

.2657  22.1728 

.5314  6.9208 

.5314  12.0048 
Mean. 

Gen.  Ellis 2  1.7677 

.2  5.8269 

.2  96381 

.1  1.1470 

.1  10.8819 

.1  17.7400 

.5  2.1516 

I  .5  9.0600 

.5  17.2650 

Weisbach |  .0328  2.0000 

|  .0656  20000 

'. I  .0984  2.0000 

n                           .1312  2.0000 

.0328  .8333 

.0656  .8333 

.0984  .8333 

.1312  .8333 
Mean. 


PRACTICAL  HYDRAULICS. 

An  inspection  of  Table  9  shows  that  the  coefficient 
of  flow  for  small  orifices  and  for  small  velocities,  is 
greater  than  it  is  for  large  orifices  and  for  great 
velocities. 

It  will  also  be  observed  that  the  results  of  experi- 
ments differ  considerably,  though  the  data  employed 
is  approximately  similar. 

Thus  Castel  finds  the  coefficient  of  flow  for  an  ori- 
fice .0328  feet  diameter,  under  a  head  of  2.132  feet,  to 
be  .673:  while  Weisbach  finds  it,  for  an  orifice  .0328 
feet  diameter,  under  a  head  of  2  feet,  to  be  .628. 

Bidone's  experiments  give,  for  circular  orifices,  x= 
0.128,  and  for  rectangular  orifices,  x=  0.1  25. 

Weisbach's  experiments  give,  for  rectangular  ori- 
fices, #=0.134. 

Weisbach,  however,  employs  for  rectangular  ori- 
fices the  mean  between  these  results  —  that  is, 


Substituting  these  values  in  Eq.  (93)  there  results, 
for  circular  orifices: 

.12871).  (94) 


And  for  rectangular  orifices: 

(95) 


PEACTICAL   HYDRAULICS.  87 


TO  FIND  THE  COEFFICIENT  OF  DISCHARGE  OF  PARTIAL 
CONTRACTION  FOR  CIRCULAR  AND  FOR  RECTANGULAR 
ORIFICES. 


Rule  28. — Case  1st — The  orifice  being  circular,  add 
1.  to  .128  times  the  ratio  of  the  entire  perimeter  to  the 
part  suppressed,  and  multiply  this  sum  by  the  coeffi- 
cient of  discharge  of  perfect  contraction. 

Case  2d — The*"  orifice  being  rectangular,  add  1  to 
0.143  times  the  ratio  of  the  entire  perimeter  to  the 
part  suppressed,  and  multiply  this  sum  by  the  coeffi- 
cient of  discharge  of  perfect  contraction. 

Ex.  49. — A  rectangular  orifice  being  1  foot  wide, 
6  inches  high,  and  the  head  10  feet,  what  is  the  coeffi- 
cient of  discharge  if  the  contraction  at  one  end  be 
suppressed? 

Gal. — By  Table  7,  coefficient  of  perfect  contraction 
for  the  given  head  and  given  orifice  is -=.601. 

Part  suppressed  — 6  inches— .5  feet. 

Entire  perimeter  —  1  +  1-J-.5  +  .5  —  3  feet. 

Ratio  of  entire  perimeter  to  paTt  suppressed  —  0/ — 
0. 143  times  this  ratio ;  0. 143  X  V  = . 024. 

Sum  of  1  and  this  product  =  1.024. 

This  sum,  multiplied  by  .601,  the  coefficient  of  per- 
fect contraction,  .60l)<1.024=- .615,  the  coefficient  of 
partial  contraction.=  ^.7is. 

Ex.  50. — A  rectangular  orifice  being  1  foot  wide, 


88  PRACTICAL    HYDRAULICS. 

6  inches  high,  and  the  head  10  feet,  what  is  the  co- 
efficient of  partial  contraction  if  the  contraction  at 
both  ends  be  suppressed? 

Gal. — By  Table  7,  the  coefficient  of  perfect  con- 
traction, for  the  given  head  and  given  orifice,  is— .601. 

Part  suppressed  6"  +  6"=12"=l  foot. 

Entire  perimeter,  1  +  1-f  .5+.5=3  feet. 

Ratio  of  entire  perimeter  to  part  suppressed=J, 
0.143  times  the  ratio;  0.143X  J=0.048. 

Sum  of  1  and  this  product=  1.048. 

This  sum,  multiplied  by  .601,  the  coefficient  of  per- 
fect contraction. 

.601X1. 048=. 630,  the  coefficient  of  partial  con- 
traction.— Ans. 


TO    DETERMINE   THE    COEFFICIENT    OF    CONTRACTION 
FOR  A  GIVEN  ORIFICE  AND  GIVEN  HEAD  OF  WATER. 


Let  a=the  hight  of  orifice;  &— the  breadth  of  ori- 
fice; ft— head  of  water. 

And  let  c,  cn,  n,  p,  pt  and  x  have  the  same  offices 
as  assigned  them  under  the  heading,  "Partial  Con- 
traction;" c6=the  coefficient  of  contraction  due  the 
breadth. 

In  Table  7,  the  hights  of  the  orifice  vary  from  4 
feet  to  0.125  feet,  while  the  breadth  of  each  orifice  is 
1  foot.  It  is  evident,  if  the  contraction  be  suppressed 
at  both  ends  of  any  orifice  given  in  Table  7,  the  con- 


PRACTICAL   HYDRAULICS.  89 

traction  due  the  horizontal  lips  only,  each  1  foot  in 
length,  will  obtain.  Now  if  the  lips  be  increased  any 
given  number  of  times  1  foot,  the  contraction  will  be 
proportionately  increased.  This  being  done,  if  the 
contraction  due  the  ends  be  restored,  and  the  result 
divided  by  the  length  of  the  elongated  orifice,  or  by 
the  given  number  of  times  that  the  lips  were  increased 
in  length,  the  quotient  will  express  the  mean  contrac- 
tion due  1  foot  breadth  of  the  given  orifice. 

For  an  orifice  in  Table  7,  whose  hight  is  a,  and 
whose  head  of  water  is  h,  if  the  contraction  of  both 
ends  be  suppressed,  the  ratio  n~~ — 2-Ka>  an(i  the  co~ 
efficient  of  partial  contraction: 


Multiplying  both  sides  of  Eq.  (96)  by  b,  the  breadth 
of  the  given  orifice,  restoring  the  end  contraction, 
2lil>  dividing  the  result  by  the  breadth  6,  substituting 
cb  for  left  hand  member,  and  reducing, 

/-.     .       2ax    \  2cax  /Q7\ 

Substituting  in  (97)  the  values  of  #=0.143,  and 


cb=c  (1  +  0  14371)— ^=s.  (98) 

Rule  29.— Find,  as  by  Rule  28,  the  value  of  the  co- 
efficient of  partial  contraction  for  an  orifice  of  the 
given  hight,  1  foot  wide,  and  having  the  contraction 
at  both  ends  suppressed. 

From  the  value  so  found  deduct  the  quotient  arising 


90  PBACTICAL  HYDRAULICS. 

from  dividing  0.143  times  the  product  of  the  coeffi- 
cient of  perfect  contraction,  and  the  ratio  of  the  entire 
perimeter  of  the  orifice  1  foot  wide,  to  the  part  sup- 
pressed, by  the  breadth,  of  the  given  orifice.  The  re- 
mainder will  be  the  coefficient  of  contraction  due  the 
given  orifice.  Rule  29  is  derived  from  formula  (98). 

Ex.  51. — A  rectangular  orifice  being  5  feet  wide,  3 
inches  high,  and  the  head  of  water  3  feet,  what  is  the 
coefficient  of  contraction? 

Gal. — By  Table  9,  the  coefficient  of  perfect  con- 
traction, for  an  orifice  1  foot  wide,  3  inches  high,  un- 
der a  head  of  8  feet,  is=.607. 

Part  suppressed=3"+3"=6"==.5  feet. 

Entire  perimeter=l-f-l-{-.5-=2.5  feet. 

Ratio  of  entire  perimeter  to  part  suppre8sed^~^fiT= 
.2;  0.143  times  this  ratio;  0.143X  .2-=  .0286. 

Sum  of  1  and  this  product=  1.0286. 

This  sum,  multiplied  by  .607,  the  coefficient  of  per- 
fect contraction,  gives  the  value  of  the  coefficient  of 
partial  contraction,  when  the  contraction  of  both 
ends  is  suppressed, 

cn=1.0286X.607=-.6244. 

By  Rule  29,  0.143  times  the  product  of  the  coeffi- 
cient of  perfect  contraction,  and  the  ratio  of  the  en- 
tire perimeter  of  the  orifice  1  foot  wide,  to  the  part 
suppressed,  divided  by  the  breadth  of  the  given  ori- 
fice, 0.l43x.2X.607-f  5=.0035; 

c6=.6244— .0035-.  621.— Ans. 


PRACTICAL    HYDRAULICS.  91 

Ex.  52. — A  rectangular  orifice  being  2  feet  wide, 
4  feet  high,  and  under  a  head  on  center  of  2.5  feet, 
what  is  the  coefficient  of  discharge? 

CaL — By  Table  7,  the  coefficient  of  perfect  con- 
traction, as  determined  by  experiment,  for  an  orifice  1 
foot  wide,  and  otherwise  conforming  to  the  given  con- 
ditions, is -=.629. 

Part  suppressed  (both  ends)  4+4—8  feet. 

Entire  perimeter,  1  +  1  +  8—10  feet. 

Ratio  of  entire  perimeter  to  part  suppressed  =^-5- 
-=.8;  0.143  times  this\ratio;  0.143X.8-0.1144. 

Sum  of  1  and  this  product=  1.1144. 

This  sum  multiplied  by  .629:  1.1144X.629  =  .70l. 

By  Rule  29, .0.143  times  the  product  of  the  coeffi- 
cient of  perfect  contraction,  and  the  ratio  of  the  entire 
perimeter  of  the  orifice  1  foot  wide,  to  the  part  sup- 
pressed, divided  by  the  breadth  of  the  given  orifice; 
0.l43x.8X.629-+-2=-0.036, 

Cb=.t  701— .036 -=.665.— Ans. 

Formulas  (95)  and  (98),  and  Rules  28  and  29,  based 
upon  the  mean  results  of  the  experiments  of  Bidone 
and  Weisbach,  give  but  approximations  to  the  true 
coefficients  sought.  They  are,  however,  sufficiently 
accurate  for  most  cases  occurring  in  practice.  Ex- 
ample 52  is  an  extreme  case.-  Yet  the  coefficient  .665, 
determined  from  its  solution,  seems  practically  correct, 
or  not  too  large,  in  presence  of  the  fact  that  the  area 
of  the  given  orifice  is  twice  as  great  as  that  of  the 
tabulated  orifice  whose  coefficient  of  discharge  is  .629; 


92  PKACTICAL    HYDBAULICS. 

while  the  perimeter  or  contracting  boundary  of  the 
former  is  to  that  of  the  latter  as  12  is  to  10.  Still  it 
is  to  be  admitted  that,  in  determining  the  coefficient 
for  a  given  orifice,  the  result  is  more  satisfactory  when 
the  hight  of  the  tabulated  orifice  employed  does  not 
much  exceed'  its  breadth. 


IMPERFECT  CONTRACTION. 


A        G 

FIG.  16. 


In  the  flow  of  a  stream  from  an  orifice,  the  head  of 
water  being  nominally  still,  and  the  orifice  small,  in 
relation  to  the  side  of  the  vessel  in  which  it  lies,  the 
contraction  is  called  perfect;  the  water  arriving  with 
considerable  velocity  at  the  orifice,  as  through  a  con- 
duit, A  G  F  E,  Fig.  16,  which  cross  section  varies 
from  1  to  20  times  that  of  the  orifice,  the  contraction 
thence  is  termed  imperfect. 

Let  c— coefficient  of  perfect  contraction;  cn— coeffi- 
cient of  imperfect  contraction;  n—  ratio  of  the  cross 
section  of  the  conduit,  A  G  F  E,  Fig.  16,  through  A  E, 
to  the  area  of  the  orifice  O;  A— area  of  orifice;  A,— 
area  cross  section  of  conduit. 


PKAOTICAL  HYDEAULICS  93 

The  values  of  imperfect  contraction  given  by  Weis- 
bach,  as  determined  by  his  experiments  and  calcula- 
tions, are: 

1st. — For  circular  orifices: 

cn=^c  [1  +  0.04564  (14.821"—  1)  ].  (99) 

2d. — For  rectangular  orifices: 

cn=c  [1  +  0.76  (9»—l)].  (100) 

Equation  (99)  for  circular  orifices  is  readily  resolved 
into  this  form : 

££1^=0.04564  (14.821"— 1),  (101 ) 

c 

And  equation  for  rectangular  orifices  into  this: 

£=^£=0.076  (9n— 1).  (102) 

C 

The  length  of  the  conduit  or  adjunct  is  assumed  to 
be  three  times  its  diameter,  or  not  sufficiently  great 
for  the  flow  of  water  to  be  sensibly  affected  by  side 
friction,  as  occurs  in  long  pipes. 

A^ 

By  giving  fractional  values  to  n=-r- ,  or  values  not 

A/ 
greater  than  1,  numerical  values  corresponding,  are 

f> Q 

found  for  the  expression  -     -  in  equations  (101)  and 

(102). 

In  illustration :  assume  the  areas  of  the  orifice  equal 


94 


PRACTICAL    HYDRAULICS. 


to  1  square  foot,  and  the  area  of  the  cross  section  of 
the  conduit,  A  G  F  E,  equal  to  2  square  feet;  then  n= 


Substituting  the  value  of  n  in  Eq.  102, 
^=^-0.076  (94—1). 


(103) 


Now  the  |  power  of  9,  in  other  words  the  square 

root  of  9=3;  3—1=2;  hence  ^=£=^ 0.076X2 -.152, 

c 

correction  found  for  n— J  —  0.5,  in  Table  2. 

In  the  computation  of  Tables  10  and  11,  different 
values  from  71-=.  05 — the  common  difference  being 
.05 — to  Ti  —  1,  are  employed. 


TABLE   10. 


Corrections  of  the  Coefficients  of  Plow  for  Circular  Orifices. 
Weisbach. 


n  

0.05 

0.10 

0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

0.45 

0.50 

-*  _ 

0.007 

0.014 

0.023 

0.034 

0.045 

0.059 

0.075 

0.092 

0.112 

0.134 

c 

H  

0.  55 

0.60 

0.65 

0.70 

0.75 

0.80 

0.85 

0.900 

0.95 

1.00 

<!,,"-€ 

01  fii 

01  SQ 

ft  99^ 

ft  9fift 

ft  W* 

Odftfi 

ft  4-71 

OKAfi 

n   «i  o 

C      ' 

If  n  has  any  value  not  found  in  Table  10  or  Table 
11,  substitute  such  value  in  equation  99  in  case  the 


PRACTICAL   HYDRAULICS. 


95 


given  orifice  is  circular,  or  in  equation  (100)  in  case 
the  orifice  is  rectangular,  and  solve  by  means  of  loga- 
rithms. 

TABLE  11. 


Corrections  of  the  Coefficients  of  Flow  for  Rectangular  Ori- 
fices.— Weisbach. 


n  

0.05 

0.10 

0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

0.45 

0.50 

cn-c 

c 

0.009 

0.019 

0.030 

0.042 

0.056 

0.071 

0.088 

0.107 

0.128 

0.152 

n  

0.55 

0.60 

0.65 

0.70 

0.75 

0.80 

0.85 

0.90 

0.95 

1.00 

CM  —  C 

C 

0.178 

0.208 

0.241 

0.278 

0.319 

0.365 

0.416 

0.473 

0.537 

0.608 

EXAMPLES  AND  CALCULATIONS  ILLUSTRATING  THE 
USE  OF  TABLES  10  AND  11. 


Ex.  53. — The  diameter  of  a  circular  orifice  being  6 
inches— .5  feet,  the  head  on  center  9.06  feet,  the  area 
of  the  orifice  one-fourth  (.25),  that  of  the  cross  sec- 
tion of  the  conduit  A  G  F  E,  Fig.  16,  what  is  the  co- 
efficient of  discharge? 

Gal.— By  Table  9,  the  coefficient  of  discharge 
through  a  circular  orifice  6  inches  diameter  — .  5  feet  in 
a  thin  partition  under  a  head  of  9.06  feet  of  water, 
nominally  still,  as  observed  by  Gen.  Ellis,  is=.  60191; 
say  c-=.602. 


96  PRACTICAL  HYDRAULICS. 

By  Table  10,  the  value  corresponding  to  n=.25,  the 
given  ratio,  is: 


c 
Solving  this  equation  for  cn, 


Substituting  the  value  of  c=.602  in  the  last  equa- 
tion, there  results: 

cM=1.045X.602=.629.—  Ans. 

Ex.  54.  —  A  rectangular  orifice  being  9  inches  high 
and  1  foot  wide  in  the  end  of  a  conduit,  as  A  G  F  E, 
Fig.  16,  1  foot  high,  1.25  feet  wide,  and  3  feet  long, 
under  a  head  of  3.5  feet  on  center,  of  water  nominally 
still  in  tank,  B  A  D  C,  what  is  the  coefficient  of  dis- 
charge ? 

By  Table  7,  the  coefficient  of  discharge  due  the 
given  orifice  and  given  head  of  water  nominally  still  is 

c=.609, 

Area  of  orifice,  .  95  X  1  =  .  95, 
Area  of  cross-section  of  conduit,  1.25X1=1.25. 
Ratio  of  transverse  sections  -ft—  ^-^=.6. 
By  Table  11,  the  value  corresponding  to  ti=.6,  the 

{*    •—-  -  f* 

ratio  of  transverse  sections,  is  —  —  =0.208;  whence 

c 

cn=1.208c. 

Substituting  in  last  equation  the  value  of  c=.609. 
<V=1.208X.609=.736.—  -Ana. 


PBACTICAL  HYDRAULICS.  97 

Ex.  55. — An  orifice  2  feet  square  in  the  end  of  a 
conduit,  AG  FE,  Fig.  16,  2.5  feet  square,  6  feet  long, 
under  a  head  of  5  feet  on  center  of  water,  nominally 
still  in  tank,  B  A  D  C,  what  is  the  discharge  in  cubic 
feet  per  second? 

Gal.  1st. — By  Table  7,  the  coefficient  of  perfect  con- 
traction applicable  to  an  orifice  1  foot  wide,  2  feet 
high,  under  a  head  of  five  feet  of  water  nominally 
still,  is  c=.612. 

By  Rules  28  and  29: 

Part  suppressed  (both  ends)  2  +  2=4  feet. 

Entire  perimeter  (tabulated  orifice)  1  -f  1  +  4=6  feet. 

Ratio  of  entire  perimeter  to  part  suppressed =f, 
0.143  times  this  ratio;  0.143X|=.0953. 

Sum  of  1  and  this  product=  1.0953. 

This  sum,  multiplied  by  tabulated  coefficient, 

1.0953X.612=.670. 

0.143  times  the  product  of  the  coefficient  of  perfect 
contraction,  and  the  ratio  of  the  entire  perimeter  of 
the  orifice  1  foot  wide  to  the  part  suppressed,  divided 
by  the  breadth  of  the  given  orifice, 

0.143X. 612X1-2=0.029. 

Coefficient  due  given  orifice  (2  feet  square)  under 
head  of  still  water: 

Cfc=.670— .029=. 641. 
Area  of  given  orifice,  2X2=4  square  feet. 


98  PRACTICAL    HYDRAULICS. 

Area  of  cross  section  of  conduit,  2.5X2.5=6.25 
square  feet. 

Ratio  of  transverse  sections,  n=^  T=.64. 

By  Table  11,  the  value  corresponding  to  71=.  6  5 
(nearest  to  .64),  the  ratio  of  transverse  sections  is 


c 

By  interpolation  between  values  corresponding  to 
•n—  .60,  and  n=.65,  there  results  approximately: 

^»    _  /i 

—  —  =.234;  whence, 


cn=1.234c. 

Substituting  the  value  of  c&=.641,  as  before  found, 
for  c  in  the  last  equation,  and  there  results: 

<V=1.234X.641=.791. 

By  Table  6,  square  root  of  head  i/5=2.236. 

By  Rule  27,  Q=.  791X8.025X4X2.236=56.78  cu- 
bic feet.  —  Ans. 

Gal.  2d.  —  By  Table  7,  the  discharge  found  for  an 
orifice  1  foot  wide,  2  feet  high,  under  a  head  of  5  feet, 
is=21.98  cubic  feet  per  second. 

If  it  be  assumed,  that  for  practical  purposes,  the 
discharge  through  the  given  orifice  2  feet  square,  in 
Example  55,  will  be  proportionate  to  the  tabulated 
discharge,  there  will  result: 

Q=21.98X2=43.96  cubic  feet  per  second. 

By  comparison,  it  is  seen  the  result  by  Cal.   1   ig 


PRACTICAL   HYDRAULICS.  99 

nearly  30  per   cent  greater   than   that   obtained   by 
Gal  2d. 

This  discrepancy  seems  to  illustrate  the  necessity  of 
careful  investigation  in  essaying  the  determination  of 
problems  of  practical  hydraulics,  however  tedious  the 
process  may  be. 


COEFFICIENT  OF  THE  FLOW  OF  WATER  THROUGH  A 
VERTICAL  RECTANGULAR  ORIFICE,  UNDER  A  HEAD  IN 
MOTION. 


FIG.  17. 

The  case  in  which  the  head  of  water  is  in  motion, 
occurs  for  the  most  part  in  open  channels.  In  Fig. 
17,  ABCD  represents  a  vertical  section  lengthwise 
of  a  stream  of  water  in  an  open  channel.  B  C  a  dam 
across  the  stream,  in  which  is  an  orifice,  E  F  in  hight. 
The  dam  is  assumed  to  act  as  a  restraint,  but  not 
sufficient  to  sensibly  affect  the  mean  velocity  of  the 
stream  of  water  above  it. 

Let  c= coefficient  of  discharge  under  a  head  of 
water,  nominally  still. 

cn= coefficient  of  discharge  under  a  head  of  water 
in  motion,  and  dependent  for  its  value  on  the  ratio  n. 


100 


PEACTIOAL   HYDKAULICS. 


A=area  of  orifice. 

Ay=area  of  cross  section  of  canal  or  channel. 

A 

n  ==—  ratio  of  these  areas,  not  exceeding  J. 
A, 

Weisbach  gives  as  the  result  of  his  experiments,  the 
head  being  measured  1  meter=3.28  feet  above  the 
dam. 

n  C  f\      s*  A  •+          "I     -*--*-       I  f\     n    A  <+  *>  f~lf\A\ 


=0.641  !  =0.641  n\ 

c  A 


Whence, 


(105) 


To  FIND  THE  COEFFICIENT  OF  DISCHAEGE  UNDER  A 
HEAD  OF  WATER  IN  MOTION. 

Rule  30. — Add  1  to  .641  times  the  square  of  the 
ratio  of  transverse  sections — that  of  the  canal  to  that 
of  the  orifice — and  multiply  this  sum  by  the  coeffi- 
cient of  discharge  due  the  given  orifice  and  given 
head  as  though  it  were  in  still  water. 

Rule  derived  from  formula  (105J. 

TABLE  12. 


Corrections  of  the  Coefficients  of  Plow  Through  Rectangular 
Orifices  Under  a  Head  of  Water  in  Motion.— Weisbach. 


n  10.05 
Cn  ~  c  0  00° 

0.10 
0.006 

0.15  [0.20  [0.25 
0.014|o.026|0.040 

0.30 
0.058 

0.35 
0.079 

0.40  (0.45 
0.1030.130 

0.50 
0.160 

c     1 

PEACTICAL  HYDKAULICS-  101 

Ex.  56.  A  dam  containing  a  rectangular  orifice  5 
feet  wide,  1  foot  high,  put  across  a  flume  6  feet  wide, 
raises  the  water  5  feet  in  hight  above  the  bottom  of 
the  flume,  and  3.5  feet  above  the  lower  edge  of  the 
orifice.  What  is  the  discharge  in  cubic  feet  per  sec- 
ond ? 

Gal.  1st. — Half  hight  of  orifice=.5  feet. 

Head  on  center  3.5 — .5=3  feet. 

Area  of  orifice  5X1=5  square  feet. 

Cross  section  of  flume,  6 X  5=30  square  feet. 

Ratio  of  transverse  sections  ^V~i- 

Square  of  ratio  (£)2=0.0278. 

By  Table  7.  Coefficient  of  perfect  contraction  for 
an  orifice  1  foot  wide,  1  foot  high,  under  a  head  of  3 
feet  is=.605. 

By  Rule  28.— Part  (both  ends)  suppressed  1  +  1=2 
feet.  Entire  perimeter  (tabulated) =4  feet.  Ratio  of  en- 
tire perimeter  to  part  suppressed,  £=.5 ;  0.143  times  this 
ratio;  .143X  .5=.0715.  This  sum  multiplied  by  .605, 
the  coefficient  of  perfect  contraction,  gives  the  value 
of  the  coefficient  of  partial  contraction  when  the  con- 
traction of  both  ends  is  suppressed. 

c.=1.0715X  .605^.648. 

By  Rule  29.— 0.143  times  the  product  of  the  coefii- 
cient  of  perfect  contraction  and  the  ratio  of  the  entire 
perimeter  of  the  orifice  1  foot  wide,  to  the  part  sup- 
pressed, divided  by  the  breadth  of  the  given  orifice; 

0.143x.5X.605-^-5=.009. 


102  PRACTICAL   HYDRAULICS. 

Whence, 

c6=.648—  .009=.639. 

Substitute  the  value  of  c&=.639  for  c,  and  the  value 
of  the  square  of  the  ratio;  (£)2=.0278  informula  (105) 
or  employ  Eule  30. 


cn=(l  +  .64lX.0278)X.639=.650. 

By  Table  6: 

Square  root  of  given  head  of  3  feet=1.732. 
By  Rule  27: 

#=.650X8.025X5xr.732=45.17  cubic  feet  per 
second.  —  Ans. 

Gal  2d.  By  Table  7:  The  discharge  found  for  an 
orifice  1  foot  wide,  1  foot  high,  under  a  head  of  3  feet, 
is  8.  41  cubic  feet  per  second.  If  it  be  assumed  that 
for  practical  purposes  the  discharge  through  the  given 
orifice,  5  feet  wide,  1  foothi^h,  in  Example  56,  will  be 
proportionate  to  the  tabulated  discharge,  there  will 
result  : 

8.41X5=42.05.—  Ans. 

A  discrepancy  of  3.12  cubic  feet,  or  7T4¥  per  cent. 
FLOW  OF  WATER  THROUGH  SHORT  TUBES. 


Short  tubes  or  adjutages  are  cylindrical,  conical  or 
compound  in  form. 

Cylindrical   Tubes. — The   length  of   a   cylindrical 


UNIVERSITY 

OF         vV  "^ 

PBAOTICAL   HYDBAULICS.  103 

tube  being  from  2.5  to  3  times  its  diameter,  the  mean 
coefficient  of  flow  through  it  as  determined  by  the  ex- 
periments of  Bidone,  Eytelwine,  D'Aubuisson  and 
Weisbach,  is  .815,  while  under  otherwise  similar  cir- 
cumstances, the  mean  coefficient  of  discharge  through 
an  orifice  in  a  thin  plate  is  .615.  The  ratio  of  .815  to 
.615  is  1.325;  that  is,  the  discharge  through  a  short 
tube  of  the  given  proportions  (2.6  to  1),  is  1.325  times 
as  much  as  the  discharge  through  an  orifice  of  equal 
diameter  in  a  thin  plate.  For  practical  purposes  this 
ratio  may  be  assumed  general  in  its  application  with- 
out material  error. 

Hence,  to  find  the  coefficient  for  a  short  tube,  hav- 
ing given  the  coefficient  of  an  orifice  in  a  thin  plate, 
of  equal  diameter,  and  under  an  equal  head. 

Rule  31. — Multiply  the  given  coefficient  of  the  ori- 
fice by  1.325. 

Ex.  57. — The  diameter  of  a  short  tube — length  to 
diameter  as  2.6  to  1 — being  6  inches,  and  the  head 
9.06  feet,  what  is  the  coefficient  of  discharge? 

CaL — Given  diameter  6"=. 5  feet. 

By  Table  9,  coefficient  due  orifice,  .5  feet  diameter, 
under  9.06  feet  head=.602. 

By  Rule  31,  . 602 X  1.325=. 798.— Ans. 

In  case  there  are  no  experiments  on  which  to  rely, 
the  mean  coefficient  .815  is  to  be  employed. 

Let  d==  diameter  in  feet  of  a  cylindrical  tube  whose 
length  is  from  2.5  to  3  times  the  diameter. 

&=head  of  water  on  center. 

c=.8l5,  coefficient  of  discharge. 


104  PEACTICAL    HYDBAULICS. 

a=.7854c?2,  area  of  cross  section  of  tube. 
(2=discharge  in  cubic  feet  per  second. 
Substituting  the  values  of  a,  c  and  h=ht  in  equa- 
tion (92). 

Q=.815X8.025X.7854dV*.  (106) 

Whence, 

Q=5.137dV*.  (107) 

To  find  the  flow  of  water  through  a  cylindrical  tube 
whose  length  is  from  2.5  to  3  times  the  diameter. 

Rule  32. — Multiply  the  square  root  of  the  head  of 
water  on  center  by  5.137  times  the  square  of  the  di- 
ameter of  the  tube. 

Rule  32  derived  from  equation  (107). 

Ex.  58. — A  tube  being  3  inches  in  diameter  and  8 
inches  long,  and  the  head  of  water  in  the  center  being 
5  feet,  what  is  the  discharge  in  cubic  feet  per  second? 

Gal.— Diameter  3"=.  25  feet. 

Square  of  diameter  .25X  .25=. 0625  square  feet. 

By  Table  6:  Square  root  of  head,  i/5=2.236. 

Q=5.137X.0625X2.236=.718  cubic  feet.=^7is. 

In  case  the  proportion  of  length  to  diameter  is 
much  changed,  as  1  to  1,  the  coefficient  of  flow  is 
nearly  the  same  as  that  for  a  thin  plate,  or  if  the 
length  be  much  increased  over  three  times  the  diame- 
ter, the  coefficient  .815  becomes  diminished  according 
to  the  occurrence  of  friction  of  the  sides  of  the 
lengthened  tube,  which  is  termed  a  pipe. 

Conical    Tubes. — Conical  tubes  are   convergent  or 

divergent.     The  outer  orifice  being   smaller  than  the 
" 


PRACTICAL   HYDRAULICS. 


105 


inner,  the  tube  is  convergent:  but  if  larger,  the  tube 
is  divergent. 

Convergent  Tubes.— Extensive  experiments  have  been 
made  by  D'  Aubuisson  and  Castel  on  the  flow  of  water 
through  convergent  tubes.  These  were  made  with 
tubes  of  various  sizes  and  proportions;  but  mostly 
with  those  .61  inches  diameter  at  the  discharging  end, 
1 .59  inches  at  the  inlet  end,  and  under  a  head  of  water 
9.84  feet.  The  results  of  their  experiments,  as  stated 
by  Weisbach,  are  given  in  the  following  table: 


TABLE  13. 


Coefficients  of  discharge  and  velocity  for  How  through 
conlcally  convergent  tubes. 


Smaller  diameter=.61  inches. 


Angle  of 
Convergence 

Coefficient 
of  Flow. 

Coefficient 
of  Velocity. 

Angle  of 
Convergence 

Coefficient 
of  Flow. 

Coefficient 
of  Velocity. 

0°  0' 

0829 

0.829 

13°  24' 

0.946 

0.963 

1°  36' 

0.866 

0.867 

14°  28' 

0941 

0.966 

3°  10' 

0895 

0.894 

16°  36' 

0.938 

0.971 

4°  10' 

0.912 

0.910 

19°  28' 

0.924 

0.970 

5°  26' 

0.924 

0.919 

21°  0' 

0.919 

0.972 

7°  52' 

0.930 

0.932 

23°  0' 

0914 

0.974 

8°  58' 

0.934 

0.942 

29°  58' 

0.895 

0.975 

10°  20' 

0.938 

0.951 

40°  20' 

0.870 

0.980 

12°  4' 

0942 

0955 

48°  0' 

0.847 

0.984 

Ex.  59.     The  smaller  diameter  of  a  conically  con- 
vergent tube  being  6  inches,  the  angle  of  convergence 


106  PEAOTICAL  HYDEAULICS: 

5°  26'  and  the  head  of  water  on  center  9.06  feet, 
what  is  the  flow  of  water  in  cubic  feet  per  second? 

Gal. — Diameter  6  inches=.5  feet. 

By  Table  9  the  coefficient  corresponding  to  the 
given  diameter  and  head=  .602,  and  coefficient  cor- 
responding to  .61  inches  on  which  Table  13  is  based 
=.618. 

By  Table  13  the  coefficient  corresponding  to  the 
given  angle  of  convergence  5°  26'  is=.924. 

Ratio  of  coefficients  .602-.618=.974. 

Then  coefficient  of  flow  due  the  given  diameter 
.924X  .974=.900. 

And  cross  section  of  tube  . 5 X. 5 X. 7854— 1963 
square  feet. 

By  Table  6,  square  root  of  head=y/9jo6=3.01 
nearly. 

By  Rule  27,  Q=.900X8.025X.  1963X3.01=4.27 
cubic  feet. — Ans. 

Divergent  Tubes. — Experiments  show  that  the  flow 
of  water  through  a  shor£  divergent  tube  is  similar  to 
that  in  a  'thin  plate,  the  coefficients  of  which  are 
given  in  Table  9.  In  ordinary  practice  .62  is  em- 
ployed. 

In  case  a  vacuum  is  formed  in  a  divergent  tube,  the 
flow  is  greatly  increased,  so  that  it  may  then  even  ex- 
ceed the  theoretical  flow  due  the  force  of  gravity, 
through  an  orifice  in  a  thin  plate,  whose  diameter  is 
equal  to  that  of  the  smaller  diameter  of  the  divergent 
tube ;  in  other  words,  its  coefficient  of  flow  becomes 
greater  than  unity.  The  conditions  effecting  this  re- 


PKACTICAL    HYDRAULICS. 


107 


suit  are  a  high  velocity  of  flow  in  a  tube  of  small  di- 
vergence, and  whose  length  is  several  times  its  smaller 
diameter.  Thus,  the  smaller  diameter  of  a  divergent 
tube  being  1.32  inches,  the  length  9  times  this  diame- 
ter:^ 1.88  inches,  the  included  angle  of  the  tube 
equal  to  5°  6',  and  the  head  of  water  2.89  feet,  Ven- 
turi  found  the  coefficient  of  flow  equal  to  1.46,  or  2.4 
times  that  of  an  equal  orifice  in  a  thin  plate.  If  the 
entry  end  of  an  otherwise  similar  tube  be  bell- 
mouthed  in  form,  the  coefficient  of  flow  estimated  for 
the  smaller  diameter  will  evidently  exceed  that  ob- 
tained by  Venturi.  The  principle  of  the  formation  of 
a  vacuum  by  flowing  water  at  a  high  rate  of  velocity, 
through  a  divergent  tube,  and  thereby  greatly  increas- 
ing the  volume  of  discharge,  was  known  to  the 
ancients.  D'Aubuisson  states  that  the  application  of 
the  principle,  at  a  distance  less  than  52.5  feet  from  the 
public  conduits  of  Rome,  by  Roman  citizens  having 
grants  of  water,  was  prohibited  by  Roman  law . 


TABLE  14. 


Coefficients  of  the  flow  of  water  through  divergent  tubes. 


Length  of 

Lenffth  of 

Angle. 

Tube. 

Coefficient. 

Angle. 

Tube. 

Coefficient. 

Feet. 

Feet. 

3°  30' 

0.364 

0.93 

5°  44' 

.193 

.82 

4°  38' 

1.095 

1.21 

i     10°  16' 

.865 

.91 

4°  38' 

1.508 

1.21 

10°  16' 

.147 

.91 

4°  38' 

1.508 

1.34 

14°  14' 

.147 

.61 

5°  44' 

0.57 

1.02 

108  PRACTICAL    HYDEAULICS. 

Ex.  60. — In  a  divergent  tube  the  smaller  diameter 
being  .61  of  an  inch,  the  length  1.508  feet,  the  angle 
included  between  its  sides  4°  38',  and  the  head  on  cen- 
ter 2.89  feet,  what  is  the  volume  of  flow  in  cubic  feet 
for  24  hours? 

Cat. — Diameter  .61  inches=.0508  feet. 

By  Table  14,  mean  coefficient  of  discharge  (1 .21  + 
1.34)^2=1.275. 

Area  of  cross  section  of  tube,  . 0508 X- 0508 X. 7854 
=  .002027  square  feet. 

By  Table  6,  square  root  of  head  1/^9= 1.7. 
By  Rule  27,  volume  of  discharge  per  second, 
Q=1.275X8.025X.002027X1.7=. 03525. 

In  24  hours  are  86,400  seconds;  hence,  .03525 X 
86400=3046.05  cubic  feet.— Ans. 

Ex.  61. — In  a  compound  tube,  Fig.  18,  the  cylin- 
drical part,  P,  is  .0853  feet  in  diameter,  2.0605  feet  in 
length;  the  convergent  part,  C,  .2559  feet  long;  the 
divergent  part,  D,  .7667  feet  in  length;  and  the  head 
2.3642  feet.  What  will  be  the  discharge  in  10  hours? 

CaL — By  Table  15,  the  coefficient  of  flow  due 
CPD=.905. 

Compound  Tubes. — Compound  tubes  are  of  various 
forms.  Eytelwine,  as  stated  by  J.  T.  Fanning,  after 
experimenting  with  cylindrical  tubes  of  uniform 
diameter  and  different  lengths,  placed  between  them 
and  the  reservoir  convergent  tubes  of  the  form  of  the 
contracted  vein,  and  renewed  the  experiments;  then 


PRACTICAL   HYDRAULICS. 


109 


added  to  the  discharge  end   a   divergent   tube   with 
5°  6'  angle.     Fig.  18. 


D 


FIG.  18. 

In  Fig.  18,  C  represents  the  conically  convergent 
part  of  the  tube  of  the  form  of  the  contracted  vein; 
P,  the  cylindrical  part  of  uniform  diameter,  but  of 
different  lengths,  and  the  conically  divergent  part 
with  5°  6'  angle.  The  results  obtained  are  given  in 
the  following  table: 

'%  < 

TABLE  15. 


Coefficients  of  the  flow  of  water  through  compound  tubes. 


Head. 
Feet. 

Diameter 
of  P 
Feet. 

Liength   of 
Pin 
Diameter. 

Length   of 
Pin 
Feet. 

Coefficient 
for  P. 

Coefficient 
forCP. 

Coefficient 
forCPD. 

2.3642 

0.0853 

0.038 

0.0033 

0.62 

2.3642 

0.0853 

1.000 

0.0853 

.62 

.967 

2.3642 

0.0853 

3.000 

0.2559 

.82 

.943 

1.107 

2.3642 

00853 

12.077 

1.0302 

.77 

.870 

.978 

2.3642 

0.0853 

24.156 

2.0605 

.73 

.803 

.905 

2.3642 

0.0853 

36.233 

3.0907 

.68 

.741 

.836 

2.3642 

0.0853 

48.272 

4.1176 

.63 

.687 

;762 

2.3642 

0.0853 

60.116 

5.1479 

.60 

.648 

.702 

110  PBACTICAL    HYDEAULICS. 

Area   of   cross-section   of  tube   P,    .0853X-0853X 
.7854=. 005761  square  feet. 

By  Table  6,  the  square  root  of  head  1/21*642= 1.5 4 
nearly. 

By  Rule  27,  Q=.905X8.025x.0057X1.54=.06367 
cubic  feet  per  second. 

In  10  hours  are  3600X10=36,000  seconds;  hence, 
.06367X36000=2292.07  cubic  feet.— Ans. 

Divergent  and  Compound  Tubes. — These  tubes  sel- 
dom find  a  place  in  practice.  The  lessons,  however, 
which  they  teach,  are  of  interest,  and  serve  to  stimu- 
late the  vigilance  of  the  engineer,  lest  irregularities 
occurring  from  design  or  otherwise,  shall  elude  his  ob- 
servation in  matters  of  importance. 


PRACTICAL  HYDRAULICS. 


Ill 


FLOW  OF  WATER  THROUGH  PIPES. 


'p^f^r 

:Tx~ft 


3)  gggZk-J* ft 


FIG.  19. 


The  flow  of  water  through  a  pipe  is  estimated  to 
begin  at  a  point  where  the  stream,  after  contraction, 
expands  so  as  to  fill  the  pipe,  as  at  F  G,  Fig.  19. 

The  part,  D  F  G;  performing  the  office  of  a  short 
tube,  is,  as  hereinbefore  shown,  from  2 .6  to  3  times  the 
diameter  of  the  pipe. 

The  total  head,  AD,  consists  of  three  parts:  AB, 
which  generates  the  velocity ;  B  C,  which  overcomes 
the  resistance  of  entry;  and  C  D,  which  overcomes  all 
resistances  in  the  pipe,  FGE.  CE  is  termed  the 
hydraulic  gradient,  and  is  such,  if  the  pipe  is  running 


112  PRACTICAL   HYDRAULICS. 

full,  the  water  will  rise  to  this  grade  through  tubes, 
as  a  b,  cd  and  ef. 

Short  and  Long  Pipes. — A  pipe,  exclusive  of  the 
tube  portion  described,  in  case  its  length  does  not  ex- 
ceed a  thousand  times  its  diameter,  is  termed  a  short 
pipe,  and  in  case  its  length  exceeds  a  thousand  times 
its  diameter,  is  termed  a  long  pipe. 

Let,  in  Fig.  19: 

h=A~D,  the  total  head. 

hv==A.l$,  the  velocity  head,  or  head  necessary  to 
generate  the  velocity  v. 

he—~B  C,  the  entry  head,  or  head  necessary  to  over- 
come the  resistances  of  entry. 

h^=hv +he= A  C,  the  inlet  head,  or  head  necessary 
to  generate  the  velocity,  v,  in  the  pipe,  and  to  over- 
come the  resistances  of  entry. 

hf=C  D,  the  head  necessary  to  overcome  the  resist- 
ances within  the  pipe. 

v=the  measured  velocity  of  discharge. 

^=the  theoretical  velocity  due  the  head  ht=A.  C. 

c=the  coefficient  of  flow  in  a  short  tube  (length  to 
diameter  as  3  to  1),  as  determined  by  experiment. 

cv=c,  the  coefficient  of  velocity,  as  the  stream, 
after  contraction,  fills  the  pipe. 

ce— the  coefficient  of  entry. 

cy— a  variable  coefficient  for  the  resistances  within 
pipes,  as  determined  by  experiment. 

c£=internal  diameter  of  pipe. 
j»=perimeter  or  internal  contour  of  pipe. 

a=area  of  cross  section  of  pipe. 


PEACTICAL   HYDRAULICS.  113 

£=length  of  pipe. 
s=sine  of  slope. 
r=hydraulic  mean  radius. 
/—amount  of  resistances  to  flow  in  the  pipe. 
w=  weight  of  water  discharged  during  the  time  of 
resistance  to  its  flow. 
Then  there  results: 


Equation  of  total  head,  h=hv  +  he  +  hf.  (108) 

Equation  of  entry  head,  he=hi  —  hv.  (109) 


By  equation  (8),  velocity  head,  ^p=—  . 

&9 

?;2 

By  equation  (8),  inlet  head,  hi=f^.  (Ill) 

49 

Equation  of  theoretical  velocity  due  inlet  head, 

*,=£  (112) 

Substituting  the  value  of  v(  of  (112)  in  (111), 


Substituting  the  values  of  ht  of  (113),  and  of  hv  of 
(110)  in  (109), 

(114J 


Putting  c.= 


|  1^1  (U5) 


114  PRACTICAL  HYDRAULICS. 

Substituting  ce  for  J  —5 — 1  j-  in  (114), 
K        J 

/>    -»12 

(116) 


The  work  performed  by  the  weight,  w,  falling  ver- 
tically by  the  force  of  gravity  through  the  distance, 
hj,  in  one  second,  is 

~F=whf,  "foot  pounds."  (117) 

Experiments  show  that  the  amount  of  resistances 
occurring  from  friction  of  the  internal  surfaces  of  a 
pipe,  and  from  other  causes,  varies  nearly  as  the  s'quare 
of  the  velocity,  v. 

Experiments  also  show  that  the  amount  of  resist- 
ances increases  directly  as  the  length,  I,  of  the  pipe, 
and  inversely  as  its  diameter,  d,  or  hydraulic  mean 

,.  a 

radius,  r=—  . 
P 

The  work  performed  by  the  force  of  gravity,  in 
overcoming  these  resistances,  so  as  to  effect  the  dis- 
charge of  the  weight,  w,  of  water  with  the  velocity, 
v,  per  second,  as  first  proposed  by  Chezy,  and  sub- 
quently  adopted  by  most  authors  on  hydraulics,  is: 


a 


in  which  cf  is  a  variable  coefficient  whose  values  are 
determined  by  experiment. 


PBACTICAL  HYDEAULICS,  115 

v2 
Substituting  the  value  of  hv=£-  of  (HO)  in  (118), 

and  equating  (117)  and  (118), 


Dividing  (119)  by  w,  V=-  (12°) 

Substituting  the  values  of  hv  of  (HO),  he  of  (116), 
and  hf  of  120,  in  (108), 

' 


Factoring  (121),  fc=|  1  +  c  +^}^- 

Transposing  (122)  with  respect  to  v, 
I       tgh       U 


a      n  (£2      d 
Hydraulic  radius,  r==~==~==' 


Substituting  I   for        of  (124)  in  (123), 

U 


,rrw^  (125) 

Factoring  (125), 

H— 1 U 

-7l  (126) 


Under  the  heading,  "flow  of  water  through  short 


116  PRACTICAL  HYDRAULICS. 

tubes,"  the  value  of  the  mean  coefficient  of  discharge 
has  been  shown  to  be  c='.815.  But  c==cw=.815;  that 
is,  the  coefficient  of  flow  at  the  inlet  orifice  of  a  short 
tube,  is  equal  to  the  coefficient  of  velocity  in  the  pipe, 
estimating  the  beginning  at  that  point,  where  the 
stream,  after  contraction,  expands  so  as  to  fill  the  pipe . 

Substituting  the  value  of  ct;=.815  in  equation  (115), 

ce=.505.  (127) 

Substituting  the  value  of  c  =.505  in  (125), 


r  J 

In  determining  the  velocity  of  flow  in  a  pipe,  whose 
length  exceeds  a  thousand  times  the  diameter,  the 
value  of  (l+ce)  (in  125),  being  small  in  comparison 

with  the  value  of  —  is  usually  omitted  as  insignifi- 

cant;* or,  more  direct,  let  equation  (120)  be  transposed 

1  » 

with  respect  to  v,  and  -  be  substituted  for  -: 


x  (129) 

cf 

Substituting  s=~,  the  sine  of   slope,    CED,    in 
(129), 


Hydraulicians  have  given  different  empirical  formu- 
las for  the  determination  of  the  values  of  cf.  Thus: 
Weisbach,  assuming  that  the  resistance  of  friction  in- 


PEACTICAL    HYDRAULICS. 


117 


creases  simultaneously  as  the  square,  and  as  the  square 
root  of  the  cube  of  the  velocity  finds  as  follows: 

3=(4  c,)=0.01439+  <L.<Li;u.5..  (131) 

V   v 

This  formula,  as  claimed  by  its  author,  agrees  more 
accurately  with  observations  than  do  those  of  the  older 
hydraulicians.  In  the  experiments  from  which  it  was 
derived,  the  velocity  varied  from  0.14  feet  to  15.25 
feet  per  second,  and  the  pipes  from  1.06  inches  to 
5.31  inches  in  diameter. 

H.  Darcy's  formula  for  velocity,  resolved  with  re- 
spect to  this  coefficient,  gives: 


This  formula  was  deduced  from  very  extensive  ex- 
periments. In.  these  the  variation,  with  respect  to 
velocity,  was  from  0.29  feet  to  16.24  feet  per  second, 
and  with  respect  to  diameters  of  pipes,  from  3  inches 
to  20  inches  nearly. 

Weisbach  remarks  of  this  formula,  that  it  "is  not 
sufficiently  accurate  for  small  velocities." 

J.  T.  Fanning's  "Series  of  Coefficients  of  Flow  (m)" 
[m— c/]  "of  water  in  clean  pipes,  under  pressure,  at 
different  velocities,  and  in  pipes  of  different  diame- 
ters"— from  which  the  following  table  is  extracted — 
seem  more  simple  and  comprehensive  on  this  subject 
than  can  well  be  rendered  in  a  single  formula.  These 
coefficients  are  deduced  from  experiments,  in  which 
the  variation  in  velocities  was  from  0.18  feet  to  46.7 
feet  per  second,  and  in  which  the  diameters  of  the 
pipes  were  from  \  inch  to  3  feet. 


118 


PRACTICAL    HYDRAULICS. 


TABLE  16. 

Coefficients  of  resistance  to  the  Plow  (cy)  of  Water  in  Clean 
Pipes.     Extracts  from  Fanning. 


Velocity. 
Feet  per 
Second. 

DIAMETERS. 

i-iuch 
Coef. 

1-inch 
Coef. 

3-inch 
Coef. 

6-inch 
Coef. 

12-inch 
Coef 

24-inch 
Coef. 

48  -inch 
Coef. 

96-inch 
Coef. 

.1 

.0150 

.0119 

.0080 

.0073 

.0067 

.2 

.0143 

.0116 

.0079 

.0072 

.0066 

.3 

.0137 

.0113 

.0078 

.0072 

.0066 

.0055 

.4 

.0133 

.0110 

.0078 

.0071 

.0065 

.0054 

.5 

.0128 

.0107 

.0077 

.0071 

.0065 

.0054 

.0040 

.6 

.0124 

.0104 

.0077 

.0070 

.0064 

.0054 

.0040 

.0029 

.7 

.0120 

.0102 

.0076 

.0070 

.0064 

.0053 

.0040 

.0029 

.8 

.0116 

.0100 

.0075 

.0069 

.0063 

.0053 

.0040 

.0029 

.9 

.0113 

.0097 

.0075 

.0069 

.0063 

.0053 

.0040 

.0029 

1.0 

.0110 

.0095 

.0074 

.0068 

.0062 

.0053 

.0040 

.0029 

1.1 

.0107 

.0093 

.0074 

.0068 

.0062 

.0052 

.0039 

.0029 

.2 

.0104 

.0091 

.0073 

.0067 

.0062 

.0052 

.0039 

.0029 

.3 

.0101 

.0090 

.0073 

.0067 

.0061 

.0052 

.0039 

.0029 

.4 

.0099 

.0088 

.0072 

.0067 

.0061 

.0051 

.0039 

.0028 

.5 

.0096 

.0087 

.0072 

.0066 

.0061 

.0051 

.0039 

.0028 

.6 

.0094 

.0085 

.0072 

.0066 

.0060 

.0051 

.0039 

.0028 

.7 

.0092 

.0084 

.0071 

.0066 

.0060 

.0051 

.0039 

.0028 

.8 

.0090 

.0083 

.0071 

.0065 

.0060 

.0051 

.0039 

.0028 

1.9 

.0088 

.0082 

.0070 

.0065 

.0060 

.0050 

.0039 

.0028 

2. 

.0086 

.0081 

.0070 

.0065 

.0059 

.0050 

.0038 

.0028 

2.25 

.0084 

.0079 

.0069 

.0064 

.0059 

.0050 

.0038 

.0028 

2.5 

.0080 

.0077 

.0068 

.0063 

.0058 

.0049 

.0038 

.0028 

2.75 

.0078 

.0075 

.0068 

.0063 

.0058 

.0049 

.0038 

.0028 

3. 

.0075 

.0073 

.0067 

.0062 

.0057 

.0048 

.0038 

.0028 

3.5 

.0073 

.0071 

.0066 

.0061 

.0056 

.0048 

.0037 

.0028 

4. 

.0072 

.0070 

.0065 

.0061 

.0055 

.0047 

.0037 

.0027 

5. 

.0070 

.0068 

.0064 

.0059 

.0054 

.0047 

0037 

.0027 

6. 

.0069 

.0067 

.0062 

.0058 

.0053 

.0046 

,0036 

.0027 

7. 

.0068 

.0066 

.0061 

.0057 

.0053 

.0045 

.0036 

.0027 

8. 

.0066 

.0065 

.0060 

.0056 

.0052 

.0045 

.0036 

.0026 

9. 

.0065 

.0064 

.0059 

.0056 

.0051 

.0045 

.0036 

.0026 

10. 

.0064 

.0063 

.0058 

.0055 

.0051 

.0044 

.0035 

.0026 

12. 

.0063 

.0061 

.0058 

.0054 

.0050 

.0044 

.0035 

.0026 

14. 

.0062 

.0061 

.0057 

.0053 

.0049 

.0043 

.0035 

.0026 

16. 

.0062 

.0060 

.0057 

.0053 

.0049 

.0043 

18. 

.0062 

.0060 

.0057 

.0053 

20. 

.0062 

.0060 

.0057 

PRACTICAL    HYDRAULICS.  119 


COMPARISON  or  THE  VALUES  OF  THE  COEFFICIENT 
OF  RESISTANCES,  Cft  AS  FOUND  BY  WEISBACH,  DARCY 
AND  FANNING. 

Ex.  62. — The  velocity,  in  a  clean  pipe  J  foot  di- 
ameter, being  1  foot,  what  is  the  coefficient  of  resist- 
ances ? 

Gal.  1st. — By  Weisbach's  formula  (131),  square  root 
of  given  velocity: 

l/v=l. 

Substituting  the  value  of  ]/v  in  (131),  z—4cc~ 
0. 01439 +iL-OJTT_i_5_5 ;  whence, 

c/=.0079.—  Ans. 

Gal.  2d.— By  Darcy's  formula  (  132),  hydraulic 
mean  radius  r=5.=1^r. 

Substituting  value  of  r  in  (132),  c  =.0066.— Ans. 

Gal.  3. — By  Table  16,  from  Fanning's  series  of  co- 
efficients, cy— .0074. — Ans. 

Ex.  63. — The  velocity  in  a  clean  pipe  2  feet  di- 
ameter, being  4  feet  per  second,  what  is  the  coefficient 
of  resistance? 

Gal.  1st. — By  Weisbach,  formula  (131),  square  root 
of  given  velocity: 

1/'U=:1/4=2. 

Substituting  value  of  i/v  in  (131),  2=4  c/^0.01439 
+  jL-_o_yjj>_5;  whence, 

Cf==.0057.— Ans. 


120  PRACTICAL  HYDRAULICS. 

Cat.  2d.—  By  Darcy's  formula  (132),  hydraulic  mean 
radius  r=J=-|=J.. 

Substituting  value  of  r  in  (132), 

c/=.0052.—  -Ans. 
Gal  3d.—  By  Table  16,  Fanning's: 
c,=.0047.  —  Ans. 

Ex.  64.  —  The  velocity  in  a  clean  pipe  4  feet  diame- 
ter, being  9  feet  per  second,  what  is  the  coefficient  of 
resistance? 

Gal.  1st.  —  By  Weisbach's  formula  (131),  square  root 
of  velocity, 

}/v=l/9=S. 

Substituting  value  of  ^v  in  (131),  2=4c/=0.01439 
5;  whence, 


c/--=.0050.—  Ans. 

Gal.  2d.  —  By  Darcy's  formula  (132),  hydraulic  mean 
radius  r=£.=£=l. 

Substituting  value  of  r  in  (132), 

c/=.0051.—  Ans. 
Gal.  3d.—  By  Table  16, 

c/=.0037.—  Ans. 

Ex.  65.  —  The  velocity  in  a  clean  pipe  8  feet  diame- 
ter, being  9  feet  per  second,  what  is  the  coefficient  of 
resistance? 


PRACTICAL   HYDRAULICS,  12X 

Cat,  1st,—  By  Weisbach's  formula  (131),  square  root 
of  velocity, 

/v=]/9:=3. 

Substituting  value  of  ]/v  in  (131),  z=^4i  ^=0.  01439 
4-jL_o_yj_5_5;  whence, 


<V-=.0050.—  Ans. 

Cal.  2d.—  By  Darcy's  formula  (132),  hydraulic  mean 
radius  r=|-=f  =2. 

Substituting  value  of  r  in  (132), 


Gal.  3d.—  By  Table  16,  from  Fanning's  series  of 
coefficients, 

c/^.0026.—  Ans. 

Weisbach  states  that  his  formula  for  tta  coefficient 
of  resistance  (z)  "is  founded  upon  the  assumption  that 
the  resistance  of  friction  increases  at  the  same  time 
with  the  square  and  with  the  square  root  of  the  cube 
of  the  velocity."  He  further  says  "that  the  values 
from  newer  experiments  show  that  the  coefficient  of 
resistance  (z)  for  the  friction  of  water  in  tubes  de- 
creases not  only  as  the  velocity  (v)  increases,  but  also, 
although  more  slowly,  as  the  diameter  of  the  pipe  be- 
comes greater."  He  omits,  however,  to  amend  his 
formula  so  as  to  embrace  this  element. 

The  coefficient  of  resistance  (c/  in  our  notation)  as 
deduced  from  Darcy's  formula  for  velocity,  decreases  as 
the  diameter  of  the  pipe  increases. 

An  inspection  of  the  results  obtained  by  Darcy's 


122  PRACTICAL   HYDEAULICS. 

formula  for  Examples  64  and  65  would  show  that  this 
diminuition  practically  ceases,  when  the  diameter  ex- 
ceeds 4  feet.  Thus,  for  a  pipe  4  feet  diameter,  the  co- 
efficient found  is  .0051,  and  for  a  pipe  8  feet  diameter, 
it  is  .0050.  The  results  for  large  pipes,  by  Darcy's 
formula,  are  not  in  conformity  with  those  obtained  by 
later  experiments. 

By  Table  16,  from  Fanning's  series,  the  coefficients 
under  the  imposed  conditions,  as  seen,  are  .0037  for  a 
4-foot  pipe,  and  .0026  for  an  8-foot  pipe. 

It  is  to  be  noticed  that,  in  general,  the  results  of 

later  experiments  with  respect  to  the  velocity  of  water 

in  large  pipes  closely  approximate  those  tabulated  by 

Mr.  Fanning  in  terms  of  the  coefficient  of  resistance. 

Thus  F.  P.  Stearns,  M.  Am.  Soc.    C.    E.,  in  a  paper 

read  before  the  American  Society  of  Civil  Engineers, 

October  1,  1884,  states  that  three  experiments  made 

with  the  "  Sudbury  Conduit,"  a  cast-iron  pipe  4  feet 

diameter  and  1747  feet  in  length,  coated  with  a  coal  tar 

preparation,  and  in  good  condition,  resulted  as  follows: 

Mean  velocity,  4.966  feet. 

Mean  coefficient  of  velocity,      142.11  feet. 

Mean  value  of  rit  0.001221  feet. 

Substituting  the  values  of  the  mean  velocity  here 
given,  i>=4.966  feet,  and  the  value  of  ri—rs~ 
0.001221,  in  equation  (130)  and  resolving  with  respect 
to  the  coefficient  of  resistance, 

Cf=Q.  003188. 
Referring  to  Table  16,  we  find  the  coefficient  of  re- 


PRACTICAL  HYDRAULICS.  123 

sistance  corresponding  to  a  velocity  of  5  feet  (nearest 
approximate  to  4.966  feet)  in  a  pipe  4  feet  diameter, 

(y=0.0037. 

Substituting  the  value  of  ^=0.0037  in  Eq.  (130), 
and  resolving  with  respect  to  the  coefficient  of  velocity, 
c,  as  employed  by  Mr.  Stearns,  there  results:  c=131.9, 
as  compared  with  142.11. 

Mr.  Stearns  further  states,  in  the  paper  referred  to, 
as  follows : 

"The  experiments  of  Hamilton  Smith,  Jr.,  M.  Am. 
Soc.  C.  E.  (transactions  for  April,  1883),  give  curves 
of  coefficients  for  pipes  up  to  30  inches  diameter.  Ex- 
tending these  curves  would  give  for  a  48 -inch  pipe, 
with  velocity  of  5  feet  per  second,  a  coefficient  of  128, 
as  compared  with  142.1  above." 

"  The  experiments  of  S.  N.  Tubbs,  M.  Am.  Soc.  C. 
E.,  on  pipes  of  2  and  3  feet  diameter,  would  give  by 
extending  the  curves  a  coefficient  about  the  same  as 
that  in  the  average  above." 

"The  experiments  of  Dr.  Lampe,  at  Danzig,  give 
results  somewhat  higher  than  those  of  Mr.  Hamilton 
Smith,  Jr." 

The  formulas  applied  to  Example  3,  to  find  the 
values  of  the  coefficient  of  resistance  for  a  3-inch  pipe, 
give  by 

Weisbach,  c/=0.0079. 

And  by  Darcy,  c,=0.0066. 

The  value  by  Fanning  is  <?/=().  0074. 

We  thus,  perceive  concerning  the  values  of  this  co- 


124  PRACTICAL   HYDRAULICS. 

efficient  for  small  pipes,  that  Fanning  differs  less  than 
7  per  cent  from  Weisbach,  whose  experiments  were 
confined,  as  hitherto  shown,  to  this  class  of  pipes, 
while  Darcy  differs  nearly  20  per  cent  from  him.  It 
must  not  be  understood,  however,  that  these  differences 
obtain  to  the  same  extent  in  estimating  the  velocity. 
Reference  to  Eq.  (130)  shows  that  the  coefficient  of 
velocity  depends  not  upon  the  direct  value  of  the  co- 
efficient of  resistance,  but  upon  the  square  root  of  it. 
So  that  in  estimating  the  velocity  or  quantity  of  flow, 
the  real  difference,  in  the  case  cited,  between  the  re- 
sults of  Weisbach  and  Fanning  would  amount  to  only 
about  3J  per  cent. 


INTERPOLATION  IN  TABLE  16. 


The  general  formula  of  Darcy  furnishes  a  simple 
means  of  interpolation  with  respect  to  the  coefficients 
of  resistance,  when  the  difference  between  the  ex- 
tremes is  not  large.  Thus  the  general  formula  is: 

(133) 


In  which  r  represents  the  mean  hydraulic  radius  m 
and  n  auxilliary  numbers  whose  values  are  determined 
by  substituting  the  tabulated  values  of  r  and  cf  in 
(133)  between  which  latter  intermediate  values  are 
sought  corresponding  to  different  values  of  r'.  The 
values  thus  found  for  m  and  n  are  employed  as  con- 


PRACTICAL    HYDRAULICS.  125 

stants  in  the   determination  of    these    intermediate 
values  of  q/>. 

For  example,  the  velocity  of  flow  being  3  feet  per 
second,  let  it  be  required  to  determine  the  values  of  c/ 
due  the  diameters  30,  36,  42  and  44  inches,  between 
24  and  48  inches. 

The  hydraulic  mean  radii  of  24  inches  and  48  inches, 
are  respectively  |=|  and  £=1.  By  Table  16,  the 
value  of  Cf  due  a  pipe  24  inches  diameter  is  .0048,  and 
that  due  a  pipe  48  inches  diameter  is  .0038.  Substi- 
tuting these  values  of  r  and  c/  in  Eq,  (133),  there  re- 
sults ; 

(134) 


and  m  +  7i=.  0038.  (135) 

Subtracting  (135)  from  (134), 

7i=.0010;  (133) 

Substituting  value  of  n  of  (136)  in  (134), 

m=.0028.  (137) 

The  hydraulic  mean  radii  corresponding  to  the 
given  intermediate  diameters,  are  respectively  f  ,  f  ,  | 
and  If 

Substituting  the  values  of  m=.0028,  n—.QQlQ,  and 
r=f  ,  |,  f  and  \\  in  (133),  there  results  for  the  given 
diameters  : 

30",  C/=.0028+  I  (.0010)=.  0044;  (138) 

36",  c/=.0028+  4  (.0010)=.  0041;  (139) 


126  PRACTICAL  HYDRAULICS: 

42",  cf=. 0028+   4  (.0010)=.0039;  (160) 

44",  c/=.0028  +  H-  (.0010)^.0039;  (141) 

to  be  interpolated  as  required. 

Thus,  by  interpolation,  may  Table  16  be  completed 
with  the  approximate  values  of  the  coefficient  (c/)  of 
resistance. 


To  FIND  THE  VELOCITY  OF  WATER  FLOWING  THROUGH 
SHORT  PIPES. 

Rule  33. — Extract  the  square  root  of  64.4  times 
the  given  head  (total  head)  in  feet,  divided  by  1.505, 
increased  by  the  product  of  the  length  of  the  pipe  in 
feet,  and  the  coefficient  of  resistance — coefficient  found 
in  and  computed  from  Table  16 — due  the  given  di- 
ameter, divided  by  the  hydraulic  mean  radius. 

Rule  33  corresponds  Eq.  (128). 

EXAMPLES  AND  CALCULATIONS  WITH  RESPECT  TO  THE 

FLOW  OF  WATER  THROUGH  SHORT  PIPES. 

Ex.  66. — A  pipe  being  1  foot  in  diameter,  10  feet 
long,  and  the  head  of  water  being  one  hundred  feet, 
what  will  be  the  velocity  of  flow  per  second? 

Gal. — As  the  pipe  is  very  short,  it  is  evident  that 
a  large  portion  of  the  head  will  be  expended  in  gener- 
ating the  velocity.  Let  it  be  assumed  that  not  less 


PEACTICAL  HYDRAULICS,  127 

than  one-half  the  total  head  will  be  so  expended.  In 
which  case  the  velocity  will  exceed  50  feet  per  sec- 
ond. Turning  to  Table  16,  we  perceive  that  the  co- 
efficient of  resistance  due  a  velocity  of  16  feet  in  a 
pipe  1  foot  diameter  is  .0049,  and  that  the  decrease  of 
the  preceding  coefficients  in  the  1  foot  diameter  col- 
umn, corresponding  to  the  increase  of  velocity,  is  very 
small.  Let,  then,  the  least  coefficient  in  column  be 
taken,  c/=.0049. 

Hydraulic  mean  radius  r=J. 

Substituting  these  values  of  c/=.0049,  r=J;  also, 
the  values  of  2^=64.4,  h=lOO  feet,  and  1=  10  feet  in 
Eq.  (128) ;  or,  in  other  words,  apply  Rule  33 : 

64.4X100 


Whence,  v=6l.53  feet. — Ans. 

It  will  be  remembered  that  equation  (128),  or  Rule 
33,  is  to  be  employed  in  finding  the  velocity  when  the 
given  length  of  the  pipe  is  less  than  one  thousand 
times  its  diameter. 

Ex.  67.—  The  head  being  10  feet,  what  will  be  the 
velocity  of  water  flowing  through  a  pipe  2  feet  in  di- 
ameter and  1000  feet  long? 

Cat. — Assume  by  way  of  trial  that  the  velocity  will 
be  9  feet  per  second. 

By  Table  16,  the  coefficient  of  resistance  due  9  feet 
velocity  in  a  2-foot  pipe  is  c/=.0045. 

Hydraulic  mean  radius  r=|=J. 

Substituting  the  values  of  c/=,0045,  r=J,  h=10, 


128  PRACTICAL  HYDRAULICS. 

£=100,  and  2</=64.4,  in  equation  (128);  or  applying 
Bule  33: 

64.4X10  U 

L.505+.0045X1000-5-JJ 

Whence,  v=7.83  feet,  trial  result. 

Turning  again  to  Table  16,  we  find  that,  corres- 
ponding to  8  feet  velocity  nearest  approximate  to  our 
trial  result  7.83  feet,  the  coefficient  of  resistance  is 
c/=,0045,  the  same  as  employed  in  our  calculation. 

Therefore  we  have: 

v=7.83  feet.— Ans. 

Had  the  tabulated  coefficient  of  resistance  corres- 
ponding to  the  velocity  found  by  trial,  not  have  been 
equal  to,  or  closely  approximate  to,  that  employed  in 
our  calculation,  it  would  have  been  necessary  to  repeat 
the  operation,  using  this  new  coefficient. 

Ex.  68.— The  head  being  20  feet,  the  pipe  4  feet 
diameter,  and  4,000  feet  long,  what  is  the  velocity  of 
flow  per  second? 

Gal.  1st. — Assume  the  velocity  for  the  purpose  of 
trial  9  feet.  Then: 

By  Table  16,  c^O. 035. 

Hydraulic  mean  radius  r=£— 1. 

Substituting  the  values  of  c/-— .0036,  r=l,  A=20 
feet,  £=4,000  feet,  and  2(/=64.4  in  Eq.  (128);  or  ap- 
plying Rule  33: 

f  64.4X20  U 

~"\1.505  +  .0036X4000-5-l  j 

Whence,  v=8.999  feet.— Ans. 


PBACTICAL   HYDRAULICS.  129 

Gal.  2. — Employing  formula  (130),  which  is  for  a 
long  pipe,  there  results: 


_(       64.4X20        } 
~  "\.0036X4000-f-lj 


Whence;  v=9A2S  feei.—Ans. 

Comparing  these  results,  the  difference  is  seen  to  be 
.429  feet  velocity  per  second.  It  appears  quite  evi- 
dent, then,  that  1.505,  the  first  term  in  the  denomina- 
tor of  the  right  hand  member  of  Eq.  (128) — an 
equation  for  velocity  of  water  in  short  pipes — cannot 
be  omitted,  where  accuracy  is  required,  even  if  the 
length  of  the  pipe,  as  in  the  given  example,  is  1,000 
times  the  diameter. 

Ex.  69. — The  head  being  12  feet,  the  pipe  15  inches 
diameter,  1,200  feet  long,  let  it  be  required  to  find: 

The  velocity  of  flow  per  second; 

The  discharge  in  cubic  feet  per  second; 

The  loss  of  head  due  velocity; 

The  loss  of  head  due  resistance  of  entry; 

The  head  expended  in  overcoming  the  resistances 
within  the  pipe; 

The  sine  of  slope; 

The  fall  per  mile  due  resistances  in  pipe ; 

And  the  total  fall  per  mile. 

Cal. — Assume  by  way  of  trial  the  velocity  of  flow= 
6  feet  per  second. 

By  Table  16,  the  coefficients  of  flow  due  a  velocity 
of  6  feet  in  a  12-inch  pipe,  are  c/=.0053;  and  in  a 
24-inch  pipe,  c/=.0046. 


130  PRACTICAL   HYDRAULICS. 

The  hydraulic  mean  radii  are  as  follows: 

Forl2",  r=J; 
And  for  24",  r=|=J. 

Substituting  the  values  of  <y=.0053,  and  of  r=J, 
in  Eq.  (133);  also,  the  values  of  c/=  .0046,  and  of  r= 
J  in  the  same  equation,  there  results: 

m+4n=.0053.  (a) 

m +  27i=.  0046.  (6) 

Whence,  7i=.00035.  (c) 

Substituting  value  of  (ri)  in  Eq.  (a), 

m=.0039.  (c?) 

Substituting  the  values  of  m  of  (d),  and  of  -n.  of  (c) 
inEq.  (133), 


The  hydraulic  mean  radius  due  15"  r=r5K. 
Substituting  this  value  of  r=T67  in  Eq.  (e), 

c/-=. 0039 +  .00035-s-T8T=.  0050.  (f) 

Substituting  the  values  of  <y=:0050,  r  =  -^j-,  ^^=12 
feet,  i  =  l,200  feet,  and  2^-64.4,  in  Eq.  (128),  or 
applying  Rule  33, 


.505+.  0050  X 1 200-5- ^J 
Whence,  ^=6.11  feet. — Ans.  (h) 


PRACTICAL   HYDRAULICS.  131 

The  value  of  v  of  Eq.  (h)  is  so  near  the  assumed 
value,  6  feet,  it  will  be  unnecessary  to  repeat  the  oper- 
ation for  finding  a  nearer  approximate  to  the  true 
value. 

Area  of  cross-section  of  given  pipe, 

a=(<-f  )2X. 7854=1.227  square  feet.  (i) 

Quantity  of  flow  is  equal  to  the  product  of  the  area 
of  cross-section  and  the  velocity;  hence, 

Q=at;=1.227x6.11=7.5  cubic  feet— Ans.  (j) 

To  find  the  loss  of  head  due  velocity,  substitute  the 
value  of  v  of  Eq.  (h),  and  the  value  of  2$ =64. 4,  in 
Eq.  (110), 

h=(l'l^=  .58  feet.— Ans.  (k) 

To  find  the  loss  of  head  due  the  resistance  of  entry, 
substitute  the  value  of  ce-=.505,  of  Eq.  (127),  and  the 
value  of  V=£-=.58,  of  Eq.  (k),  in  (116), 

he=.5SX  .505-=. 29  feet.—  Ans.  (I) 

To  find  the  head  expended  in  overcoming  the  re- 
sistances within  the  pipe,  substitute  the  values  of  hv= 
.58  of  Eq.  (k),  of  he=.2S  of  Eq.  (I),  and  of  h= 
12  feet,  the  total  or  given  head  in  Eq.  (108),  and 
transposing: 

^=12— .58— .29  =  11.13  feet.— Ans.  (m) 

To  find  the  sine  of  slope,  divide  the  head,  hf,  ex- 
pended in  overcoming  the  resistances  in  the  pipe  by 
the  length  of  the  pipe.  Thus,  the  sine  of  the  angle  of 
slope,  C  E  D,  Fig.  19,  is  in  the  given  example: 


132  PRACTICAL    HYDRAULICS. 

8=-^=ftjj=-.  009275.— Ans.  (n) 


To  find  the  fall  per  mile: 

Let  F— fall  in  feet  per  mile: 

1  mile=5280  feet.  (o) 

Then  1 :  .009275 : :  5280 :  F.  (p) 

Whence,  F  =-48.972  feet.—  Ans.  (q) 

When  1.505,  the  leading  term  in  the  denominator 
of  Eq.  (128),  or,  in  other  words,  when  Eq.  (130)  is  em- 
ployed instead  of  Eq.  (128),  it  is  assuming  that  in 
Fig.  19,  A  E  is  the  slope  or  hydraulic  gradient  instead 
of  C  E,  which  is  erroneous. 

But  when  the  pipe  is  very  long  in  comparison  with 
its  diameter,  the  error  is  insignificant. 

Let  Fm— the  entire  fall  per  mile. 

Substitute  the  values  of  /^— .58  of  Eq.  (&),  and  he  — 
.29  of  Eq.  (I)  in  Eq.  (109), 

Then  Fw=F+^48.972+.8T=49.842 feet.— Ans. 

FLOW  OF  WATER  IN  LONG  PIPES. 

For  a  given  or  determined  velocity,  the  inlet  head 
hi,  remains  constant,  regardless  of  the  length  of  the 
pipe. 

In  general,  if  F  denote  the  fall  per  mile — fall  re- 


PEACTICAL   HYDKAULICS.  133 

quisite  to  overcome  the  resistances  within  the  pipe  — 
(n)  the  number  of  miles,  length  of  the  pipe,  and  Ft  the 
total  fall,  then  will 


In  the  computation  of  the  following  table  for  the 
velocity  and  quantity  of  flow  in  long  pipes,  Eq.  (130), 


- 


in   which  the  value  of  s—-  is  given,  has  been  em- 

ployed.    The  inlet   head,   hit  if  required,  will  be  de- 
termined from  the  velocity. 


134 


PRACTICAL    HYDRAULICS. 


TABLE  17. 


Velocities  and  Quantities  of  Flow  in  Clean  Iron   Pipes  due 
given  Slopes  and  Diameters. 


Sine 

DIAMETERS. 

Fall  per 

of 

Slope 

§"=.03125  feet. 

^"=.04167  ft. 

f"=.0625  ft. 

1"=.0833  ft. 

Mile. 

h 

Velo'y 

Cubic 

Velo'y 

Cubic 

Velo'y 

Cubic 

Velo'y 

Cubic 

Feet. 

«=  f 

Ft  per 

Feet  per 

?t.  per 

Ft.  per 

Ft.  per 

Ft.  per 

Ft.  per 

Ft.  per 

t 

Sec. 

Sec. 

Sec. 

Sec. 

Sec. 

Sec. 

Sec. 

Sec. 

21.12 

.004 

26.40 

.005 

31.68 

.006 

36.96 

.007 

42.24 

.008 

.04 

.0057 

47.52 

.009 

.13 

.0062 

52.80 

.010 

1.03 

.0032 

24 

.0068 

63.36 

.012 

0.698 

0.00054 

0.89 

.0012 

1.14 

.0035 

.43 

.0078 

73.92 

.014 

0.763 

0.00059 

0.91 

.0013 

1.23 

.0038 

.54 

.0084 

84.48 

.016 

0.826 

0.00063 

0.99 

.0014 

1.34 

.0041 

.64 

.0089 

95.04 

.018 

0.888 

0.00069 

1.05 

.0014 

1.45 

.0045 

.76 

.0096 

105.6 

.02 

0.947 

0.00073 

1.10 

.0015 

1.52 

.0047 

.81 

.0099 

158.4 

.03 

1.18 

0  00090 

1.44 

.0020 

1.92 

.0059 

2.28 

.0125 

211.2 

.04 

1.38 

0.0011 

1.77 

.0024 

2.30 

.0071 

2.73 

.0149 

264.0 

.05 

1.58 

0.0013 

2.04 

.0028 

2.60 

.0080 

3.05 

.0167 

316.8 

.06 

1.79 

0.0014 

2.31 

.0032 

2.85 

.0087 

3.40 

.0186 

369.6 

.07 

1.98 

0.0015 

2.49 

.0034 

3.10 

.0095 

3.64 

.0199 

422.4 

.08 

2.13 

0.0016 

2.68 

.0037 

3.30 

.0101 

3.92 

.0214 

475.2 

.09 

2.36 

0.0018 

2.85 

.0039 

3.54 

.0109 

4.18 

.0228 

528.0 

.10 

2.49 

0.0019 

3.04 

.0041 

3.73 

.0114 

4.44 

.0242 

633.0 

.12 

2.76 

0.0021 

3.32 

.0045 

4.18 

.0128 

4.90 

.0268 

739.2 

.14 

3.04 

0.0023 

3.46 

.0047 

4.50 

.0138 

5.29 

.0289 

844.0 

.16 

3.30 

0.0025 

3.84 

.0052 

4.83 

.0148 

5.64 

.0308 

950.4 

.18 

3.50 

0.0027 

4.09 

.0056 

5.11 

.0157 

6.00 

.0328 

1050. 

.20 

3.71 

0.0028 

4.31 

.0059 

5.40 

.0166 

6.33 

.0346 

1320. 

.25 

4.15 

0.0032 

4.83 

.0066 

6.10 

.0187 

7.14 

.0390 

1584. 

.30 

4.58 

0.0035 

5.36 

.0073 

6.73 

.0206 

7.90 

.0432 

2112. 

.40 

5.32 

0.0041 

6.26 

.0086 

7.79 

.0239 

9.13 

.0499 

2640. 

.50 

5.99 

0.0048 

7.07 

.0097 

8.82 

.0271 

10.34 

.0565 

3168. 

.60 

6.62 

0.0051 

7.80 

.0107 

9.79 

.0300 

11.57 

.0632 

3696. 

.70 

7.20 

0.0055 

8.47 

.0116 

10.76 

.0330 

12.71 

.0694 

4224. 

.80 

7  75 

0.0059 

9.14 

.0125 

11.72 

.0357 

4752. 

.90 

8.22 

0.0063 

9.80 

.0134 

12.60 

.0379 

5280. 

1.00 

8.74 

0.0067 

10.39 

.0142 

PRACTICAL  HYDRAULICS. 


135 


TABLE  17. 


Velocities  and  Quantities  of  Flow  in  Clean  Iron  Pipes   due 
given  Slopes  and  Diameters. 


Fall  per 
Mile. 
Feet. 

Sine 
of 
Slope 

_i 

DIAMETERS. 

li"=.125  feet. 

If  "=.1458  ft. 

2"=.  1667  ft. 

2i"=.4167  ft. 

Velo' 
Ft  per 
Sec. 

Cubic 
Feet  per 
Sec. 

Velo'y 
Ft.  per 
Sec. 

Cubic 
Ft.  per 
Sec. 

Velo'y 
Ft.  per 
Sec. 

Cubic 
Ft.  per 
Sec. 

Velo'y 
Ft.  per 
Sec. 

Cubic 
Ft.  per 
Sec. 

21   12 

.004 

1.18 

.0258 

.35 

,0460 

26.40 

005 

.21 

.0201 

1  34 

.0292 

.52 

0518 

3L68 

!ooe 

1.19 

.0146 

]36 

!0227 

li-60 

]0327 

.68 

!0573 

36.96 

.007 

1.29 

.0158 

.45 

.0243 

1.60 

.0349 

.81 

.0617 

42.24 

.008 

1.39 

.0171 

.58 

.0264 

1.73 

.0378 

.04 

.0662 

47.52 

.009 

1.48 

.0182 

.70 

.0284 

1.87 

.0408 

2.07 

.0706 

52.80 

.010 

1.60 

.0196 

.79 

.0299 

1.98 

.0432 

2.20 

.0750 

63.36 

.012 

1.73 

.0212 

.95 

.0326 

2.22 

.0484 

2.44 

.0832 

73.92 

.014 

1.86 

.0228 

2.13 

.0356 

2.36 

.0515 

2.62 

.0003 

84.48 

.016 

2.01 

.0247 

2.22 

.0371 

2.50 

.0546 

2.78 

.0948 

95.04 

.018 

2.10 

.0258 

2.35 

•0392 

2.63 

.0574 

2.93 

.0999 

105.6 

.02 

2.28 

.0279 

2.53 

.0422 

2.80 

.0611 

3.15 

.1074 

158.4 

.03 

2.81 

.0346 

3.10 

.0518 

3.39 

.0740 

3.81 

.1299 

211.2 

.04 

3.37 

.0413 

3.69 

.0617 

4.00 

.0873 

4.51 

.1538 

264.0 

.05 

3.73 

.0458 

4.28 

.0715 

5.02 

.1095 

5.34 

.1821 

316.8 

.06 

4.11 

.0504 

4.69 

.0783 

5.50 

.  1200 

5.85 

.1995 

369.6 

.07 

4.42 

.0542 

5.02 

.0838 

5.90 

.1288 

6.34 

.2162 

422.4 

.08 

4.73 

.0580 

5.36 

.0895 

6.30 

.1375 

6.77 

.2309 

475.2 

.09 

5.05 

.0619 

5.63 

.0940 

6.61 

.1442 

7.19 

.2442 

528.0 

.10 

5.48 

.0672 

6.01 

.1003 

6.98 

.  1523 

7.61 

.2595 

633.0 

.12 

6.03 

.0740 

6.65 

.1110 

7.49 

.1634 

8.27 

.2820 

739  2 

.14 

6.54 

.0802 

7.19 

,1200 

8.01 

.1748 

8.89 

.3031 

840.0 

.16 

7.01 

.0862 

7.70 

.1285 

8.50 

.1855 

9.48 

.3233 

950.0 

.18 

7.50 

.0923 

8.22 

.1372 

8.96 

.1955 

10.04 

.3424 

1056. 

.20 

7.88- 

.0969 

8.69 

.1450 

9.38 

.2047 

10.63 

.3624 

1320. 

.25 

8.77 

.1079 

9.69 

.1617 

10.43 

.2276 

11.86 

.4054 

1584. 

.30 

9.65 

.1187 

10.62 

.1773 

11.38 

.2483 

13.15 

.4084 

2112. 

.40 

11.23 

.1380 

12.28 

.2050 

12.98 

.2833 

2640. 

.50 

12.60 

.1550 

136 


PRACTICAL  HYDRAULICS. 


TABLE  17. 

Velocities  and  Quantities  of  Flow  in  Clean  Iron  Pipes  due 
given  Slopes  and  Diameters. 


Sine 

DIAMETERS. 

Fall  per 

of 

Mile 

••? 

3"=.  25  feet. 

4"  =.333  ft. 

6"  =.5  feet. 

8"  =.667  Feet. 

Feet. 

4 

Veloc'y 
Ft.    per 
Sec. 

Cubic 
Ft.  per 
Sec. 

Velo'y 
Ft  per 
Sec. 

Cubic 
Ft  per 
Sec. 

Velo'y 
Ft  per 
Sec. 

Cubic 
Ft  per 
Sec. 

Velo'y 
Ft  per 
Sec. 

Cubic 
Ft.  per 
Sec. 

8  448 

.0016 

64 

573 

8.976 

.0017 

.68 

.586 

9  504 

.0018 

1  44 

.282 

75 

611 

10  032 

.0019 

1  48 

290 

80 

628 

10  560 

0020 

1  25 

1091 

1  52 

298 

83 

639 

11  616 

.0022 

1  31 

1144 

1  60 

314 

93 

659 

12.672 

.0024 

1.10 

.0540 

1.36 

.1187 

1.68 

.330 

2.02 

.703 

13.728 

.0026 

1.16 

.0569 

1.42 

.1235 

1.76 

.346 

2.11 

.737 

14784 

.0028 

1.22 

.0599 

1.48 

.1298 

1  83 

.359 

2.22]    .768 

15.840 

.0030 

.28 

.0630 

1.53 

.1335 

1.92 

.377 

2.32|    .808 

18.480 

.0035 

.41 

.0692 

1.68 

.1465 

2.10 

.395 

2.511    .876 

21.120 

.004 

.53 

.0749 

1.79 

.1562 

2.26 

.444 

2.69     .931 

26.40 

.005 

.71 

.0839 

203 

.1771 

2.53 

.496 

1  2.99  1.015 

31.68 

.006 

.86 

.0915 

2.20 

.1923 

2.79 

.548 

3.32^  1.157 

36.96 

.007 

2.02 

.0992 

2.46  .2146 

2.79 

.589 

[  3.32  1.262 

42.24 

.008 

2.16 

.1060 

2.67  .2339 

3.00 

.631 

3.62!  1.344 

47.52 

.009 

228 

.1119 

2.82 

.2460 

322 

.672 

1  3.85   1.424 

52.80 

.010 

2.43 

.1190 

2.96 

.2582 

3.431  .721 

1  4.08   1.496 

63.36 

.012 

2.68 

.1313 

3  23!.  2893 

3.67 

.784 

4.29 

1.614 

73.92 

.014 

2.88 

.1413 

3.48 

.3036 

3.99 

.858 

4.71 

1.782 

84.48 

.016 

3.07 

.1507 

3.71 

.3237 

4.37 

.922 

5.11 

1.916 

94.04 

.018 

3.24 

.1593 

3.91 

.3412 

4.70 

.975 

5.49 

2.033 

105.6 

.02 

3.50 

.1717 

413.3607 

497 

1.022 

5.83  2.155 

158.4 

.03 

4.24 

.2081 

5.16 

.4502 

521 

1.263 

6.181  2.667 

211.2 

.04 

5.03 

.2469 

6.11 

5331 

6.43 

1.484 

7.64|  3.145 

264.0 

.05 

5.67 

.2785 

6.83 

.5954 

7.56 

1.665 

9.01 

3.513 

316.8 

.06 

621 

,3049 

7.32 

.6390 

8.48 

1.929 

10.06 

3.847 

369.0 

.07 

6.79 

.3331 

7.99 

.6967 

9.83 

1.976 

[11.02  4.196 

4220 

.08 

7.25 

.3559 

8.60 

.7506 

10.062.114 

12  02'.. 

4752 

.09 

7.78 

.3816 

9.13 

.7960 

10.922.274 

5280 

.10 

8.24 

.4043 

9.70 

.8467 

11.582  399 

633.6 

12 

905 

.4440 

10.63 

.9270 

12.21 

739  2 

.14 

9  77 

.4977 

11.53 

1.006 

844.8 

.16 

10.46 

.5131 

1238 

1.081 

950.4 

.18 

11.08 

.5436 

1056. 

.20 

11.88 

.5832 

1320. 

.25 

13.29 

.6523 

PKACTICAL   HYDRAULICS. 


137 


TABLE  17. 


Velocities  and  Quantities  of  Plow  in   Clean  Iron  Pipes   due 
given  Slopes  and  Diameters. 


Sine 

DlAMETBRS. 

Fall  per 

of 
Slope. 

9"=.75  feet. 

10"=.  833  ft. 

11"=.  917  ft. 

"=1  foot. 

Mile. 

h 

Vdoci'y 

Cubic 

Velo'y 

Cubic 

Velo'y 

Cubic 

Veloci'y 

Cubic 

Feet. 

-i 

Feet  per 
Sec. 

Feet  per 
Sec. 

Ft  per 

Sec. 

Ft  per 

Sec. 

Ft  per 
Sec. 

Ft  per 
Sec. 

Feet 
Per  Sec. 

Ft  per 
Sec. 

4.752 

.0009 

1.54 

1.210 

•5.280 

.0010 

1.61 

1.265 

5.808 

[0011 

1.62 

1.069 

1.68 

.319 

6.336 

.'0012 

1.61 

.878 

1.70 

1.122 

1.79 

.402 

6  864 

.0013 

1.69 

.922 

1.79 

1.181 

1.88 

.476 

7.392 

.0014 

1.66 

.735 

1.76 

.960 

1.85 

1.221 

1.94 

.489 

7.920 

.0015 

1.72 

.761 

1.84 

.984 

1.92 

1.267 

2.01 

.579 

8.448 

.0016 

1.78 

.788 

1.92 

1.047 

2.00 

1.320 

2.08 

.634 

8.976 

0017 

1.83 

.810 

1.99 

1.085 

2.07 

1.366 

2.15 

.689 

9.504 

.0018 

1.89 

.836 

2.04 

1.110 

2.12 

1.399 

2.20 

.728 

10.032 

.0019 

1.96 

.867 

2.13 

1.162 

2.20 

1.452 

2.27 

.783 

10.560 

.0020 

2.01 

.890 

2.19 

1.194 

2.26 

1.492 

2.33 

1.826 

11.616 

.0022 

2.12 

.938 

2.32 

1.265 

2.39 

1.577 

2.47 

1.940 

12.672 

.0024 

2.22 

.982 

2.43 

1.325 

2.50 

1.650 

2.58 

2.026 

13.728 

.0026 

2.32 

1.025 

2.53 

1.377 

2.61 

1.723 

2.70 

2.117 

14.784 

.0028 

2.41 

1.065 

2.61 

1.423 

2.71 

1.789 

2.81 

2.207 

15.840 

.0030 

250 

1.105 

2.70 

1.470 

2.81 

1.855 

2.93 

2.297 

18.480 

.0035 

2.71 

1.198 

2.91 

1.587 

3.02 

1.993 

3.14 

2.466 

21.120 

.004 

2.88 

1.273 

3.09 

1.683 

3.24 

2.138 

3.39 

2.662 

26.400 

.005 

320 

1.414 

3.42 

1.865 

3.63 

2.176 

3.85 

3.020 

31.680 

.006 

3.54 

1.565 

3.78 

2.059 

4.00 

2.640 

4.22 

3.310 

36.960 

.007 

3.84 

1.697 

4.08 

2.222 

4.33 

2.858 

4.58 

3.601 

42.240 

008 

4.11 

1.816 

4.37 

2.383 

4.64 

3.062 

4.91 

3.856 

47.520 

.009 

4.34 

1.918 

4.61 

2.514 

4.90 

3.234 

5.19 

4.072 

52.80 

010 

4.58 

2.024 

4.88 

2.662 

5.18 

3.419 

5.48 

4.305 

63.36 

012 

5.04 

2.228 

5.38 

2.932 

5.70 

3.762 

6.02 

4.728 

73.92 

014 

5.50 

2.431 

5.89 

3.210 

6.18 

4.079 

6.49 

5.094 

84.48 

016 

5.91 

2.612 

6.33 

3.450 

6.65 

4.389 

6.98 

5.482 

95.04 

018 

6.29 

2.780 

6.75 

3.679 

7.09 

4.676 

7.44 

5.838 

105.6 

02 

6.62 

2.926 

7.07 

3.856 

7.45 

4.917 

7.84 

6.100 

158.4 

03 

8.18 

3.616 

8.73 

4.762 

9.22 

6.085 

9.72 

7.630 

211.2 

04 

9.60 

4.243 

10.20 

5.563 

10.74 

7.088 

12.28 

8.860 

264.0 

05 

10.81 

4.778 

12.29 

6.704 

12.49 

8.243 

12.69 

9.967 

316.8 

06 

11.85 

5.238 

138 


PRACTICAL   HYDRAULICS. 


TABLE  17. 

Velocities  and  Quantities  of  Flow  in  Clean  Iron   Pipes,  due 
Given  Slopes  and  Diameters. 


Sine 

DIAMETERS. 

Fall  per 

of 

Milp 

Slope. 

T 

14"=1.167   Feet- 

15"=1.25   Ft. 

16"=1.333Ft. 

!  18"=  1.5  Feet. 

Julie* 

Feet. 

'=7 

Veloci'y 
Feet  per 
Sec. 

Cubic 
Ft.  per 
Sec. 

Velo'y 
Ft  per 
Sec. 

Cubic 
Ft  per 
Sec. 

Velo'y 
Ft  per 
Sec. 

Cubic 
Ft  per 
Sec. 

(Veloc'y 
Ft.  per 
Sec. 

Cubic 
Ft  per 
Sec. 

2.112 

.0004 

1.46 

2.61 

2.640 

.0005 

1.56 

2.79 

3.168 

.0006 

1.50 

2.04 

1.66 

2.97 

3.696 

.0007 

1.56 

1.91 

1.61 

2.25 

1.76 

3  10 

4.224 

.0008 

.61 

1.71 

1.67 

2.05 

1.74 

2.43 

1.85 

3.27 

4.752 

.0009 

.71 

1.83 

1.78 

2.19 

1.86 

2.59 

1.98 

3.49 

5.280 

.0010 

.80 

1.91 

1.87 

2.30 

1.95 

2.72 

2.07 

3.66 

5.808 

.0011 

.90 

2.02 

1.98 

2.43 

2.07 

2.88 

2.20 

3.88 

6.336 

.0012 

.98 

2.11 

2.07 

2.54 

2.16 

3.02 

2.30 

4.06 

6.864 

.0013 

2.04 

2.18 

2.16 

2.65 

2.28 

3.18 

2.40 

4.23 

7.392 

.0014 

2.13 

2.27 

2.24 

2.75 

2.35 

3.28 

2.49 

440 

7.920 

.0015 

2.20 

2.35 

2.31 

284 

2.43 

3.39 

2.61 

4.61 

8.448 

.0016 

2.29 

2.44 

2.39 

2.94 

2.50 

3.49 

2.69 

4.75 

8.976 

.0017 

2.38 

2.54 

2.48 

3.08 

2.59 

3.62 

2.78 

4.90 

9.504 

.0018 

2.43 

2.59 

2.53 

3.11 

2.64 

3.69 

2.85 

5.03 

10.032 

.0019 

2.50 

2.67 

2.61 

3.21 

2.73 

3.81 

2.93 

5.17 

10.560 

.0020 

2.55 

2.72 

2.68 

3.29 

2.81 

3.92 

3.00 

5.30 

11.616 

.0022 

2.70 

2.88 

2.82 

3.47 

2.95 

4.12 

3.19 

5.63 

12.672 

.0024 

2.83 

3.02 

2.96 

3.63 

3.10 

4.32 

3.32 

5.87 

13.728 

.0026 

2.95 

3.15 

3.09 

3.79 

3.23 

4.51 

3.50 

6.18 

14.784 

.0028 

3.08 

3.29 

3.22 

3.95 

3.36 

4.68 

361 

6.38 

15.840 

.0030 

3.20 

3.42 

3.34 

4.11 

3.49 

4.87 

3.76 

6.64 

18.48 

.0035 

3.47 

3.62 

3.63 

4.46 

3.80 

5.31 

4.06 

7.17 

21.12 

.004 

3.74 

3.99 

3.90 

4.78 

4.06 

5.67 

4.33 

7.65 

26.40 

.005 

4.18 

4.46 

4.38 

5.37 

4.58 

6.39 

4.90 

8.66 

31.68 

.006 

4.60 

4.91 

4.81 

5.91 

5.03 

7.02 

5.40 

9.54 

3696 

.007 

5.03 

5.37 

5.26 

6.45 

5.49 

7.66 

5.84 

10.33 

42.24 

.008 

5.40 

5.77 

5.62 

6.90 

5.85 

8.16 

6.28 

11.09 

47.52 

.009 

5.73 

6.11 

5.95 

7.31 

6.19 

8.64 

6.63 

11.71 

52.80 

.010 

6.03 

644 

6.27 

7.70 

6.52 

9.10 

7.00 

12.37 

63.36 

.012 

6.56 

7.00 

6.84 

8.39 

7.12 

9.95 

7.73 

13.65 

73.92 

.014 

7.12 

7.60 

7.45J  9.15 

7.79 

10.87 

8.35 

14.75 

84.48 

.016 

7-66 

8.17 

7.99!  9.81 

8.33 

11.63 

8.97 

1584 

95.04 

.018 

847 

8.93 

8.03  10.47 

8.90 

12.43 

9.57 

1690 

105.6 

.02 

8.67 

9.26 

9.04  11.09 

9.41 

13.14 

10.10 

17.85 

158.4 

.03 

10.69 

11.39 

11.1313.66 

11.58 

16.17 

12.37 

21.86 

211.2 

.04 

12.38 

13.22 

12.91  15.84 

13.45 

18.77 

PRACTICAL   HYDRAULICS. 


139 


TABLE  17. 

Velocities  and  Quantities  of  Flow  in  Clean  Iron  Pipes  due 
given  Slopes  and  Diameters. 


Fall  per 
Mile. 
Feet. 

Sine 
of 
Slope. 

h 

•4 

DIAMETERS. 

20"=  1.667  feet. 

22"=1.833  feet. 

24"=2feet. 

27"=  2.  25  feet 

Veloci'y 
Feet  per 
Sec 

Cubic 
Feet  per 
Sec. 

Veloci'y 
Feet  per 
Sec. 

Cubic 
Feet  pei 
Sec. 

Velo'y 
Ft  per 
Sec. 

Cubic 
Ft  per 
Sec. 

Velo'y 
Ft  per 
Sec. 

2.00 

Cubic 
Ft  per 
Sec. 

2.112 

.0004 

7.95 

2  640 

.0005 

1.78 

5  59 

2.08 

8.27 

3.168 

.0006 

1.66 

3.61 

1.80 

4.61 

1.94 

6.10 

2.10 

8.51 

3.696 

.0007 

1.86 

4.07 

1.99 

5.25 

2.12 

6.64 

2.29 

8.91 

4.224 

.0008 

2.00 

4.35 

2.13 

5.62 

2.27 

7.13 

2.39 

9.30 

4.752 

.0009 

2.15 

4.68 

2.28 

6.01 

2.41 

7.56 

2.58 

10.26 

5.280 

.0010 

226 

4.92 

2.39 

6.32 

2.53 

7.95 

2.70 

10.74 

5.808 

.0011 

2.36 

5.15 

2.51 

6.62 

2.66 

8.34 

2.88 

11.45 

6.336 

.0012 

2.48 

5.40 

2.63 

6.94 

2.79 

8.75 

3.00 

11.93 

6.864 

.0013 

2.58 

5.62 

2.74 

7.24 

2.91 

9.14 

3.16 

12.54 

7.392 

.0014 

2.68 

5.82 

2.85 

7.51 

3.02 

9.47 

3.26 

12.96 

7.920 

.0015 

2.78 

6.05 

2.95 

7.78 

3.12 

9.80 

3.40 

13.49 

8.448 

.0016 

2.88 

6.27 

3.05 

8.05 

3.23 

10.13 

3.52 

13.98 

8.976 

.0017 

2.97 

6.48 

3.17 

8,36 

3.37 

10.57 

3.63 

14.41 

9.504 

.0018 

3.05 

6.65 

324 

8.55 

3.43 

10.77 

3.73 

14.81 

10.032 

.0019 

3.17 

6.92 

3.35 

8.85 

3.54 

11.10 

3.83 

15.21 

10.560 

.0020 

3.23 

7.05 

3.43 

9.07 

3.64 

11.43 

3.93 

15.63 

11.616 

.0022 

3.40 

7.42 

3.62 

9.55 

3.84 

12.05 

4.14 

16.44 

12.672 

.0024 

3.57 

7.79 

3.79 

10.01 

4.02 

1261 

4.34 

17.23 

13.728 

.0026 

3.73 

8.14 

3.97 

10.48 

4.21 

13.23 

4.53 

18.01 

14.784 

.0028 

3.89 

8.48 

4.14 

10.91 

4.39 

13.79 

4.72 

18.75 

15.84 

.0030 

4.02 

8.77 

4.28 

11.29 

4.54 

14.25 

4.91 

19.50 

18.48 

.0035 

4.35 

9.49 

4.64 

12.25 

4.94 

15.50 

5.32 

21.13 

21.12 

.004 

4.66 

10.16 

4.97 

13.12 

5.29 

16.62 

5.69 

22.62 

26.40 

.005 

5.24 

11.43 

5.60 

14.78 

5.96 

18.71 

6.37 

25.34 

31.68 

.006 

5.77 

12.59 

6.10 

16.20 

6.50 

20.42 

6.98 

27.74 

36.96 

.007 

6.26 

13.66 

6.64 

17.53 

7.02 

22.05 

7.52 

29.96 

42.24 

.008 

6.72 

14.66 

7.12 

18.78 

7.52 

23.61 

8.05 

31.99 

47.52 

,009 

7.13 

15.54 

7.55 

19.93 

7.98 

25.07 

8.55 

33.97 

52.80 

.010 

7.55 

16.47 

7.98 

21.06 

8.41 

26.42 

9.03 

35.89 

63.36 

.012 

8.25 

17.99 

8.74 

23.07 

9.24 

29.03 

10.00 

39.76 

73.92 

.014 

8.94 

19.49 

9.48 

2468 

10.03 

31.49 

10.87 

43.22 

84.48 

.016 

9.64 

21.03 

10.21 

26.97 

10.79 

33.90 

11.72 

46.57 

95.04 

.018 

10.30 

22.45 

10.91 

29.70 

11.52 

36.18 

12.09 

48.06 

105.6 

.02 

10.80 

23.56 

11.52 

31.15 

12.24 

38  45 

158.4 

.03 

12.23 

28.86 

12.59 

33.21 

211.2 

.04 

13.50 

2958 

140 


PRACTICAL  HYDRAULICS. 


TABLE  IT. 


Velocities  and  Quantities  of  Flow  in  Clean  Iron  Pipes,  due 
Slopes  and  Diameters. 


Fall  per 

Sine 
of 
Slope. 

DIAMETERS. 

30"=2.5  Feet. 

1=2.75  Ft. 

36"=3.  Feet. 

40"=3.333  Ft. 

i  e. 

h 

Veloci'yi  Cubic 

o'yl  Cubic 

Velo'y 

Cubic 

Velo'y  1  Cubic 

Feet. 

8—  — 

Feet  per  Feet  pe 

pei 

Ft  per 

Ft  per 

Ft  per 

Ft  Per 

Ft.  per 

I 

Sec. 

i    Sec. 

«. 

Sec. 

Sec. 

Sec. 

Sec. 

Sec. 

1.056 

.0002 

1.34 

6.58 

45 

8.61 

1.46 

10.29 

1.59 

13.88 

1.584 

.0003 

1.59 

7.78 

68 

10.21 

1.80 

12.70 

1.95 

17.00 

2.112 

.0004 

1.83 

8.99 

96 

11.65 

206 

14.56 

2.26 

19.68 

2.640 

.0005 

2.09 

10.24 

18 

12.92 

2.31 

16.35 

2.53 

22.08 

3.168 

.0006 

224 

10.97 

36 

13.99 

255 

18.02 

280 

24.43 

3.696 

.0007 

2.43 

11.90 

55 

15.14 

2.80 

19.76 

3.01 

26.27 

4.224 

.0008 

2.62 

12.84 

76 

16.36 

2.95 

20.85 

3.23 

28.14 

4.752 

.0009 

2.75 

13.48 

96 

17.58 

3.16 

22.30 

3.42 

29.80 

5.280 

.0010 

2.90 

14.21 

16  18.74 

3.32 

23.47 

3.61 

31.46 

5.808 

.0011 

3.07 

15.05 

29  19.54 

3.53 

24.91 

3.81 

33.25 

6.336 

.0012 

3.22 

1581 

4220.28 

3.70 

26.12 

3.98 

34.68 

6.864 

.0013 

3.36 

16.47 

59121  29 

3.85 

27.20 

4.15 

36.21 

7.392 

.0014 

3.50 

17.18 

7022.20 

4.00 

28.24 

4.31 

37.57 

7.920 

.0015 

3.66 

17.94 

.  ..8823.01 

4.13 

29.19 

4.49 

39.18 

8.448 

.0016 

3.79 

18.58 

4.0023.76 

4.29 

30.29 

4.65 

40.54 

8.976 

0017 

3.92 

19.21 

4.12J24.47 

4.45 

31.42 

4.80 

41.88 

9.504 

0018 

4.01 

19.66 

4.2525.22 

4.60 

32.48 

4.94 

43.07 

10.032 

0019 

4.14 

20.32 

4.40 

26.14 

4.73 

33.40 

5.08 

44.28 

10.560 

.0020 

4.24 

20.79 

4.54 

26.94 

4.88 

34.49 

5.18 

45.20 

11.616 

.0022 

4.45 

21.80 

4.76 

28.27 

5.12 

36.15 

5.52 

48.12 

12672 

.0024 

4.65 

22.83 

5.0029.02 

5.34 

37.74 

5.79 

50.48 

13.728 

0026 

4.88 

23.93 

5.23 

31.06 

5.58 

39.40 

6.04 

52.67 

14.784 

0028 

5.07 

24.86 

5.44 

32.28 

5.78 

40.86 

6.31 

55,04 

15.84 

0030 

5.27 

25.87 

5.66 

33.62 

5.98 

42.28 

6.46 

56.33 

18.48 

0035 

5.70 

27.96 

6.09 

36  17 

6.50 

45.95 

7.00 

61.09 

21.12 

004 

6.08 

29.84 

6.49 

38.57 

6.91 

48.83 

7.50 

65.41 

26.40 

005 

6.84 

33.55 

726 

43.12 

7.77 

54.89 

8.38 

73.09 

31.68 

006 

7.50 

36.79 

7.98147.40 

8.48 

59.95 

9.21 

80.32 

36.96 

007 

8.08 

39.66 

8.65J51.35 

9.22 

65.17 

9.94 

86:70 

42.24 

008 

8.64 

42.39 

9.2554.91 

9.88 

69.80 

10.61 

92.58 

47.52 

009 

9.22 

45.23 

9.8058.20 

10.52 

74.33 

11.23 

9800 

52.80 

010 

9.72 

47.71 

10.3861.62 

11.10 

78.46 

11.92 

104.00 

63.36 

012 

10.78 

52.91 

11.4568.00 

11.72 

82.84 

73.92 

014 

11.75 

57.65 

12.45173.95 

PRACTICAL  HYDRAULICS' 


141 


TABLE  17. 


Velocities  and  Quantities  of  Flow  in  Clean  Iron  Pipes  due 
Slopes  and  Diameters.       *  " 


Fall  per 
Mile. 

Sine 
of 
Slope. 
k 
f 

DIAMETERS. 

44"=3.607  feet. 

4S"=4  feet 

54"=4.5  feet. 

eo"=5  feet. 

Veloci'y 

Cubic 

Velo'y 

Cubic 

Velo'y 

Cubic 

Velo'y 

Cubic 

Feet. 

8=   '- 
I 

Feet  per 
Sec. 

Feet  per 
Sec. 

Ft  per 
Sec. 

Ft  per 
Sec. 

Ft  per 
Sec. 

Ft  per 
tec. 

Ft  per 
Sec. 

Feet  per 
Sec. 

0.528 

.0001 

1.22 

12.89 

1.28 

16.08 

1.38 

21.96 

1.52 

29.77 

1.056 

.0002 

1.72 

18.15 

1.83 

2298 

1.95 

30.97 

208 

40.84 

1.584 

.0003 

2.10 

2222 

222 

27.89 

2.42  38.53 

265 

52.09 

2  112 

.0004 

2.42 

25.55 

2.62 

32.93 

2.8445.12 

3.01 

59.04 

2.640 

.0005 

2.74 

28.87 

2.95 

37.00 

3.16J50.23 

3.44 

67.56 

3.168 

.0006 

2.98 

31.46 

320 

40.21 

3.49i55.51 

3.79 

74.32 

3.696 

.0007 

3.27 

3447 

3.48 

4367 

3.79160.21 

4.10 

80.51 

4.224 

.0008 

3.51 

37.05 

3.73 

46.81 

4.00163.61 

4.40 

86.30 

4.752 

.0009 

3.70 

39.01 

3.90 

49.06 

4.  24!  67.  20 

4.69 

91.99 

5.280 

.0010 

3.89 

41.06 

4.15 

52.15 

4.5517237 

4.94 

96.98 

5.808 

.0011 

408 

42.09 

4.38 

5495 

4  76J75.71 

5.22 

102.4 

6.336 

.0012 

426 

4497 

4.57 

57.36 

4.98179.13 

5.47 

107.3 

6.864 

.0013 

4.43 

46.77 

4.78 

60.07 

5.1982.54 

5.68 

115.5 

7.392 

,0014 

4.63 

48.83 

4.94 

62.02 

5.40  85.90 

594 

116.5 

7.920 

.0015 

4.80 

50.62 

5.13 

6447 

5.63J89.52 

6.10 

119.7 

8.448 

.0016 

4.97 

52.46 

5.30 

66.53 

5.82i92.47 

6.30 

123.7 

8.976 

.0017 

5.12 

54.04 

5.45 

68.50 

6.0095.35 

6.50 

127.6 

9.504 

.0018 

5.26 

55.48 

5.62 

70.62 

6.1497.65 

6.69 

131.3 

10.032 

.0019 

5.40 

57.01 

5.79 

72.75 

6.30J100.2 

6.87 

134.8 

10.560 

.0020 

5.58 

5885 

5.92 

74.44 

6.53 

103.8 

7.07 

138.8 

11.616 

.0022 

585 

61.71 

6.23 

78.29 

6.84 

108.8 

7.44 

146.0 

12.672 

.0024 

6.10 

6435 

6.50 

81.68 

7.14 

113.5 

7.77 

1526 

13.728 

.0026 

634 

66.87 

6.78 

85.20 

7.45 

118.5 

8.08 

158.7 

14.784 

.0028 

6.59 

69  57 

7.04 

8846 

7.74 

123.1 

838 

164.5 

15.84 

.0030 

6.85 

72.32 

7.30 

91.73 

8.06  128.2 

8.68 

1704 

18.48 

.0035 

7.98 

77.95 

7.99 

100.4 

8.741138.9 

9.37 

184.0 

21.12 

.004 

7.92 

83.60 

8.43 

105.9 

9.30 

147.9 

10.06 

197.5 

26.40 

.005 

8.85 

93.37 

950 

119.3 

10.43 

165.8 

11.30 

220.0 

31.68 

.006 

9.78 

103.3 

10.42 

130.9 

11.47 

182.4 

12.44 

244.3 

36.96 

.007 

10.59 

111.7 

11.31 

142.1 

12.45 

190.0 

4224 

.008 

11.36 

119.9 

12.25 

153.9 



47.52 

.009 

12.15 

128.3 



142 


PRACTICAL   HYDRAULICS. 


TABLE  17. 


Velocities  and  Quantities  of  Flow  in  Clean  Iron  Pipes,  due 
Slopes  and  Diameters. 


Fall  per 

•»*•?!„ 

Sine 
of 
Slope. 

DIAMETERS. 

72"=6  feet. 

84"=7  feet. 

96"  =8  feet. 

120  =10  feet. 

M  Ie. 

fl 

f 

Velcci'y 

Cubic 

Veloc'y 

Cubic 

Vel'cy 

Cubic 

Vel'cy 

Cubic 

Feet. 

s=L 

?eetper 

Feet  per 

Ft  per 

Ft  per 

Ft  per 

Ft  per 

Ft  per 

Ft  per 

i 

Sec. 

Sec. 

Sec. 

Sec. 

Sec. 

Sec. 

Sec. 

Sec. 

0.528 

.0001 

1.66 

4699 

1.91 

75.43 

214 

107.8 

2.52 

198.1 

1.056 

.0002 

2.04 

57.65 

2.72 

104.6 

3.03 

152.5 

3.65 

286.5 

1.584 

0003 

2.92 

82.53 

3.28 

126.2 

3.75 

188.5 

4.49 

352.3 

2.112 

.0004! 

3.40 

95.99 

3.78 

145.4 

4.35 

218.8 

5.20 

408.5 

2.640 

.0005 

387 

109.4 

4.23 

1628 

4.88 

245.3 

5.80 

458.7 

3.168 

.0006 

4.30 

121  6 

4.60 

177.0 

5.32 

267.4 

6.41 

503.6 

3.696 

.0007 

4.67 

132.0 

4.99 

1920 

5.78 

290.5 

6.94 

545.1 

4.224 

.0008 

4.95 

140.0 

5.40 

207.8 

6.19 

310.9 

7.42 

582.7 

4.752 

.0009 

526 

1487 

5.78 

222.4 

6.60 

324.2 

7.87 

618.1 

5.280 

.0010 

5.58 

157.8 

6.11 

235.1 

6.97 

350.5 

8.30 

651.4 

5.808 

.0011 

5.87 

166.0 

658 

253.3 

7.29 

366.2 

8.70 

6832 

6.336 

.0012 

6.12 

173.0 

6.88 

264.8 

7.60 

382.0 

9.09 

713.5 

6.864 

.0013 

634 

179.3 

7.15 

275.2 

7.92 

397.9 

9.48 

744.3 

7.392 

.0014 

66.3 

1875 

7.48 

287.7 

8.25 

414.7 

9.84 

772.5 

7.920 

.0015 

6.86 

1939 

7.70 

296.4 

8.51 

427.8 

10,18 

799.6 

8.448 

.0016 

7.08 

2002 

8.00 

307.9 

882 

443.1 

10.51 

825.7 

8.976 

0017 

7.30 

206.4 

8.22 

316.2 

9.10 

457.4 

10.84 

850.4 

9.504 

.0018 

7.50 

212  1 

8.49 

3267 

9.36 

470.5 

11.15 

875.9 

10.302 

.0019 

7.70 

217.7 

8.73 

335.8 

9.58 

481.5 

11.45 

899.9 

10.560 

.0020 

7.97 

225.2 

905 

348.3 

9.88 

496.4 

11.78 

925.3 

11.616 

.0022 

8.33 

235.5 

9.48 

364.9 

10.40 

522.8 

12.35 

971.4 

12.672 

.0024 

8.72 

246.4 

9.88 

389.1 

10.89 

547.4 

12.90 

1013.6 

13.728 

.0026 

9.06 

256.2 

10.28 

394.4 

11.34 

570.0 

13.43 

1057.4 

14.784 

.0028 

9.45 

267.2 

10.61 

408.4 

11.78 

592.1 

13.94 

1094.8 

15.84 

.0030 

9.83 

277.9 

11.00 

423.4 

12.18 

612.0 

18.48 

.0035 

10.62 

299.7 

12.55 

4830 

21.12 

.004 

11.34 

320.7 

26.40 

.005 

1268 

358.5 

PRACTICAL  HYDRAULICS.  143 


APPLICATION  OF  TABLE  17. 


Ex.  70. — The  diameter  of  a  pipe  being  4  feet,  and 
the  fall  per  mile  5.28  feet,  what  is  the  discharge  in 
cubic  feet  per  second? 

Gal— In  "fall  per  mile"  column,  Table  17,  find  5.28 
feet,  opposite  which,  in  right  hand  column,  "4t  feet 
diameter,"  will  be  found  52.15  cubic  feet,  the  dis- 
charge sought. 

Ex.  71. — The  fall  per  mile  being  10.56  feet,  a  pipe 
of  what  diameter  will  be  requisite  to  discharge  100 
cubic  feet  per  second? 

Gal.— ID.  "fall. per  mile"  column,  Table  17,  find 
10.56  feet,  opposite  which  find  100  cubic  feet,  or  near- 
est approximate  thereto:  103.8  cubic  feet  is  found  in 
right  hand  column,  headed  4.5  feet  diameter,  which, 
in  practice,  is  sufficiently  near  the  diameter  sought  in 
most  cases. 

Ex.  72. — The  flow  of  30  cubic  feet  of  water  per 
second  through  a  pipe  2.5  feet  diameter,  5  miles  long, 
is  employed  at  a  hydraulic  mine,  whose  elevation  is  400 
feet  below  the  elevation  of  the  inlet  end  of  the  pipe. 
What  is  the  effective  head  of  the  water  at  the  mine? 

Gal.— In  "2.5  feet"  diameter  column,  Table  17,  find 
29.84  cubic  feet  nearest  approximate  to  the  given  flow 
of  30  feet,  opposite  which,  in  the  "fall  per  mile"  col- 


144  PRACTICAL   HYDKAULICS. 

umn,  is  found  21.12  feet,  the  loss  of  head  per  mile. 
The  loss  in  5  miles,  the  given  length  of  pipe,  will  be: 

21. 12X5=105.60  feet. 

400—105.60=294.4  feet,  effective  head—  Ans. 

Ex.  73.— The  data  being  the  same  as  in  Ex.  72,  ex- 
cept that  the  diameter  of  the  pipe  is  3  feet  instead  of 
2.5,  what  is  the  effective  head  of  the  water  *at  the 
mine? 

CaL— Find  in  "3  feet,"  right  hand  diameter  column, 
Table  17,  30.29  cubic  feet,  nearest  approximate  to  30 
cubic  feet,  the  given  quantity,  opposite  which,  in  "fall 
per  mile"  column,  is  found  4.448  feet,  the  loss  per  mile. 

Then  8.448x5=42.24  feet  loss  of  head  in  5  miles; 
400—42.24=357.76  feet  effective  head.=4ws. 

Comparing  results  with  respect  to  the  2.5-foot  pipe 
of  Ex.  72,  and  the  3-foot  of  Ex.  73: 

357.76—294.40=62.36  feet,  it  is  seen  that  the  loss 
of  head  in  the  3-foot  pipe  is  63.36  feet  less  than  in  the 
2.5-foot  pipe,  which,  in  matters  of  economy,  is  of  no 
little  importance. 

Ex.  74. — The  elevation  of  the  Guenoc  Reservoir  Site 
of  the  Feather  River  Water  Co.  is  1015  feet  above  the 
city  base  of  San  Francisco.  The  measured  distance  of 
pipe-line  between  the  reservoir  and  the  city  is  104.83 
miles.  How  many  gallons  of  water  a  day  (24  hours) 
will  a  pipe  40  inches  diameter,  whose  inlet  is  at  the 
reservoir,  and  outlet  at*  San  Francisco,  deliver  at  an 
elevation  of  350  feet  above  city  base? 

Gal.— 1015— 350=665  feet, -total  fall;  665-^104.83 


PRACTICAL    HYDRAULICS.  145 

=6.34  feet  fall  per  mile.  In  "fall  per  mile"  column, 
Table  17,  the  nearest  approximate  fall  to  6.34  feet  is 
6.336  feet,  opposite  which,  in  "40  inches,"  right  hand 
column  "diameters,"  is  found  34.68  cubic  feet,  the  dis- 
charge per  second. 

24X60X60  =-86,400  seconds  in  24  hours;  34.68X 
86,400=2,996,352  cubic  feet  discharge  in  24  hours. 

In  1  cubic  foot  are  7.5  gallons  nearly;  2,996,352X 
7.5=22,472,640  gallons.—  Ans. 

Ex.  75. — At  a  quartz  mill,  requiring  225  effective 
horse-power,  the  efficiency  of  the  water  wheel,  in 
excess  of  the  loss  by  nozzle  resistance,  is  60  per  cent; 
the  length  of  the  pipe  line  is  3  miles,  and  the  eleva- 
tion of  the  reservoir,  at  point  of  water  supply,  is  500 
feet  above  the  point  of  application  of  the  water  at  the 
mill.  Required  the  diameter  of  the  pipe  to  carry  the 
requisite  quantity  of  water? 

Col.  1st.—  225-^.60  =  375,  total  horse-power. 

375X550=206,250  "footpounds;"  206,250-^-62.5  = 
3300  cubic  feetXl  foot,  which  we  will  term  foot  vol- 
ume. ^ 

Assume  the  loss  of  head  by  the  internal  surface 
resistances  of  pipe  10.56  feet  per  mile;  then  10.56X3 
=  31.68  feet,  total  loss  of  head. 

500—31.68=468.32  feet,  effective  fall;  3300+- 
468.32=7.05  cubic  feet  flow  required  per  second. 

In  "fall  per  mile"  column,  Table  17,  find  10.56  feet, 
opposite  which,  in  right  hand  column,  "diameters,"  find 
7.05  cubic  feet.  This  is  found  under  heading  "20 


146  PEACTICAL  HYDRAULICS. 

inches."  Hence  the  diameter  of  the  required  pipe  is 
20  inches. — Ans. 

Gal.  2d. — By  Cal.  1st,  the  "foot  volume,"  corres- 
ponding to  the  "foot  pounds,"  required  at  the  mill,  is  = 
3300  cubic  feetX  1  foot. 

Assume  4.224  feet,  the  loss  of  head  per  mile  clue 
pipe  resistances;  then  4.224X3 -=12.672,  total  loss  of 
head;  500— 12.672=487.328  feet,  effective  fall;  3300 
-^487.328—6.77  cubic  feet  required  per  second. 

In  "fall  per  mile"  column  find  4.224  feet,  opposite 
which,  in  right  hand  column,  "diameters,"  find  7.32 
cubic  feet.  This  is  found  under  heading  "24  inches," 
Hence  the  diameter  sought  is— 24  inches. — Ans. 

Cal.  3d.— By  Cal.  1st,  the  "foot  volume,"  corres- 
ponding to  the  "foot  pounds,"  required  at  the  mill,  is 
=  3300  cubic  feet  XI  foot. 

Assume  21.12  feet,  the  loss  of  head  per  mile  due 
pipe  resistances;  then  21.12X3  =  63.36  feet,  total  loss 
of  head;  500— 63.36=436.64  feet,  effective  head;  3300 
-^-436.64^=7.55  cubic  feet  required  per  second. 

In  fall  per  mile  column  find  21.12  feet,  opposite 
which,  in  right  hand  column,  "diameters,"  is  found 
7.65  cubic  feet — a  very  near  approximate  to  7.55 
cubic  feet,  the  required  quantity.  This  is  found  under 
heading  "18  inches."  Hence  the  diameter  sought  is— 
18  inches. — Ans. 

Maximum  Work. — To  determine  the  maximum 
work,  which  water  subjected  to  flow  through  a  long 
pipe  under  pressure  will  perform,  on  issuing  from  the 
pipe: 


PEACTICAL   HYDRAULICS.  147 

Let  ^=the  total  head,  exclusive  of  the  inlet  head, 
which,  rarely  in  practice  exceeds  1.5  feet. 

x—hf,  the  friction  head,  expended  in  overcoming 
the  resistances  within  the  pipe. 

d—  diameter  of  pipe. 

I—  length  of  pipe. 

<r=  £.,  hydraulic  mean  radius  regarded  constant. 

cy—  coefficient  of  friction  of  resistances  within  the 
pipe  regarded  constant. 

g—  acceleration  of  gravity. 

v—  velocity  of  water  in  the  pipe  per  second. 

n—  ratio  of  circumference  to  diameter  of  pipe. 

ht—  -h  —  x,  head  for  effective  work. 

W=  total  weight  of  water  discharged  per  second. 

w—  weight  of  a  cubic  foot  of  water. 

n<i2  „  .         _    . 

a—  -j-,  area  01  cross  section  of  the  pipe. 

u—  maximum  work  performed  by  the  water  per 
second.  Then, 

u^Wh^avwhr  (142) 

Substituting  in  (142)  ,  the  values  of 


=  —  and  h,—h  —  x\ 
4 


0f(Eq. 


x     (h-x).       (143) 
4 

Differentiating  (142),  omitting  the  constant  factors 
n,  d,  $)  Cf  I,  2  and  4, 


148  PEACTICAL   HYDBAULICS. 


Transposing  and  reducing  (144), 

»= 
Differentiating  (144), 


As  the  second  differential  coefficient  is  negative,  the 
function  u  representing  the  work  performed  is  a  max- 
imum where  #=•£,  as  found  in  Eq.  (145). 

Thus  it  is  shown  that,  in  theory,  the  work  performed 
by  water  after  flowing  through  a  pipe,  is  a  maximum 
when  the  head  expended  in  overcoming  the  resistances 
of  the  pipe  is  equal  to  one-third  of  the  total  head,  ex- 
clusive of  the  inlet  head.  A  modification  occurs,  how- 
ever, with  respect  to  this  result,  owing  to  the  experi- 
mental coefficient  of  resistance  being  variable  by 
undetermined  law,  instead  of  constant,  as  assumed  in 
our  solution.  Another  source  of  variation  is  evidently 
due  to  difference  in  diameters  of  pipes.  An  inspection 
of  Table  17  shows,  that  in  practice  the  ratio  of  the 
head  expended  in  overcoming  the  resistances  in  a  clean 
iron  pipe  is  approximately  equal  to  f  ,  as  a  mean,  in- 
stead of  J,  as  found,  of  the  total  head.  The  remain- 
ing part  of  the  total  head  amounting  to  nearly  f  as  a 
mean,  applies  to  effective  work. 

Col.  4th.—  ByCal.  1st,  the  "foot  volume"  (substitu- 
ted for  '  '  foot  pounds  "  for  convenience  of  calculation) 
required  at  the  mill,  is  3300  cubic  feet  X  1  foot. 


PRACTICAL   HYDRAULICS.  149 

Applying  the  ratio,  f ,  proposed  in  the  preceding 
article,  there  results:  500 X f— 187.5  feet,  loss  of  head 
by  friction;  1875-^. 3=62.5  feet,  loss  of  head  per  mile. 

In  "  fall  per  mile"  column,  the  nearest  approximate 
to  62.5  feet  is  63.36  feet,  loss  of  head  per  mile;  then 
63.36X3=190.08  feet,  total  loss  of  head;  500—190.08 
=309.92  feet  effective  head;  3300-^309.92=10.65 
cubic  feet  flow  required  per  second. 

Opposite  63.36  feet  in  "fall  per  mile  column," 
Table  17,  is  found  13.65  cubic  feet,  nearest  approxi- 
mate to  10.65  cubic  feet,  the  required  quantity.  This 
is  found  in  "  18  inches  "  column  of  contents  for  "  diam- 
eters." Hence  the  pipe  meeting  the  requirements 
is=l8  inches  diameter. — Ans. 

Remark. — In  "16-inch  column,"  opposite  6 3. 36  feet 
'  'fall  per  mile,"  is  found  9.95  cubic  feet,  which  is  less 
than  the  required  quantity.  Hence  a  16-inch  pipe  is 
too  small.  The  velocity  in  a  17-inch  pipe  is  v= 
(.^H^X .012X11)^=7.46  feet  per  second  for  63.36 
feet  fall  per  mile.  The  discharge  per  second  in  a  17- 
inch  pipe  will  be  (1J)2X.7854X7.46=11.76  cubic  feet 
per  second. 

This  amount,  11.76,  exceeds  the  required  amount  of 
10.65  cubic  feet.  Hence  a  17-inch  pipe  will  carry 
sufficient  water  to  do  the  work.  The  margin  of  safety, 
11.76 — 10.65=1.11  cubic  feet,  however,  is  small. 

An  18-inch  pipe  affording  a  margin  of  safety  of 
13.65 — 10.65=3  cubic  feet,  seems  by  no  means  large. 
Even  a  20-inch  pipe,  with  a  fall  of  63.36  feet  per 
mile,  and  affording  a  margin  of  7.34  cubic  feet  per 


150  PRACTICAL   HYDRAULICS. 

second,  would  not  exceed  the  limits  imposed  by 
D'Aubuisson,  in  his  advice  to  engineers,  if,  indeed,  it 
would  the  limits  of  true  economy. 

Reverting  to  the  results  obtained  by  1st,  2nd  and 
3rd  calculations  for  Ex.  75 ,  and  to  the  tabulated  re- 
sults corresponding  respectively  to  these,  or  approxi- 
mately so,  it  will  be  seen  that  in  the  first  case  there  is 
no  margin  of  safety,  in  the  second  .55  cubic  feet,  and 
in  the  third  .10  cubic  feet  per  second.  This  in 
practice  would  be  inadmissible.  A  margin  of  33  per 
cent  is  none  too  large.  So  that  if  on  portions  of  the 
lines  steeper  grades  could  not  be  had,  it  would  be  bet- 
ter to  employ  a  "  22-inch"  pipe  in  the  first  case,  a 
"  27-inch"  in  the  second,  and  a  "20-inch"  in  the 
third  case. 

Inlet  Head. — The  inlet  head  is  equal  to  the  sum  of 
the  head  expended  in  generating  the  velocity  in  a  pipe, 
and  the  head  expended  in  overcoming  the  resistance 
of  entry. 

Thus  transposing  Eq.  (109),  we  have  the  inlet  head, 

(147) 


V* 

Substituting  the  values  of  hv= —  of  Eq.  (110),  and 

c7 

of    he=ce  v*  of  Eq.  (116),  noting  that  ce=.5Q5  of  Eq. 
(127), 

i~~  '     %g~ 

Rule  34.— The  inlet  head  is  equal  to  .0234  times  the 
square  of  the  velocity  in  the  pipe. 


PBACTIG1&  HYDRAULICS, 


151 


TABLE  18. 

Velocities  in  Pipes  and  Corresponding  Inlet  Heads. 


Veloci'y 
Feet. 

Inlet 
Head. 

Feet 

Velocity. 
Feet 

Inlet 
Head. 
Feet. 

Velocity 
Feet. 

Inlet 
Head. 
Feet. 

Velocity 
Feet. 

Inlet 
Head. 
Feet. 

.80 

.015 

3.68 

.316 

6.32 

.933 

10.00 

2.34 

.90 

.019 

3.76 

.331 

6.37 

.948 

10.50 

2.58 

.00 

.023 

3.85 

.346 

6.42 

.963 

11.00 

2.83 

.13 

.030 

3.93 

.361 

6.47 

.978 

11.50 

3.10 

.27 

.038 

4.00 

.374 

6.52 

.993 

12.00 

3.37 

.39 

.045 

4.09 

.391 

6.57 

1.01 

12.50 

3.66 

.50 

.053 

4.17 

.406 

6.61 

1.02 

13.00 

3.96 

.60 

.060 

4.25 

.421 

6.66 

1.04 

13.50 

4.27 

.70 

.068 

4.32 

.436 

6.71 

1.05 

14.00 

4.59 

.79 

.075 

4.39 

.452 

6.76 

1.07 

14.50 

4.92 

.88 

.083 

4.47 

.467 

6.81 

1.08 

15.00 

5.27 

1.97 

.090 

4.54 

.482 

6.86 

1.10 

15.50 

5.62 

2.00 

.094 

4.61 

.497 

6.91 

1.11 

16.00 

5.99 

2.04 

.098 

4.68 

.512 

6.95 

1.13 

16.50 

6.38 

2.12 

.105 

4.75 

.527 

7.00 

1.15 

17.00 

6.76 

2.20 

.113 

4.81 

.542 

7.04 

1.16 

17.50 

7.17 

2.27 

.120 

4.87 

.557 

7.09 

1.17 

18.00 

7.58 

2.34 

.128 

4.94 

.572 

7.13 

1.19 

18.50 

7.79 

2.41 

.135 

5.00 

.585 

7.18 

1.20 

19.00 

8.45 

2.47 

.143 

5.07 

.606 

7.22 

1.22 

19.50 

8.90 

2.54 

.150 

5.14 

.617 

7.26 

1.23 

20.00 

9.36 

2.60 

.158 

5.20 

.632 

7.31 

1.25 

20.50 

9.83 

2.66 

.166 

5.26 

.647 

7.35 

1.26 

21.00 

10.32 

2.72 

.173 

5.32 

.662 

7.40 

1.28 

21.50 

10.82 

2.78 

.181 

5.38 

.677 

7.44 

1.29 

22.00 

11.33 

2.84 

.188 

5.44 

.692 

7.48 

1.31 

22.50 

11.85 

2.89 

.196 

5.50 

.707 

7.53 

1.32 

23.00 

12.38 

2.95 

.202 

5.56 

.722 

7.57 

1.34 

23.50 

12.92 

3.00 

.211 

5.62 

.737 

7.61 

1.35 

24.00 

13.48 

3.05 

.218 

5.67 

.753 

7.65 

1.37 

24.50 

14.05 

3.11 

.226 

5.73 

.768 

7.70 

1.38 

25.00 

14.63 

3.16 

.232 

5.79 

.783 

7.74 

1.40 

25.50 

15.22 

3.21 

.241 

5.85 

.798 

7.78 

1.41 

26.00 

15.82 

3.26 

.248 

5.90 

.813 

7.82 

1.43 

27.00 

17.05 

3.31 

.256 

5.95 

.828 

7.86 

1.44 

28.00 

18.35 

3.36 

.263 

6.00 

.843 

7.90 

1.46 

29.00 

19.78 

3.40 

.271 

6.06 

.858 

7.94 

1,47 

30.00 

21.06 

3.45 

-278 

6.11 

.873 

S.OO 

1.50 

35.00 

28.67 

3.50 

.286 

6.17 

.888 

8.50 

1.69 

40.00 

37.34 

3.55 

.293 

6.22 

.903 

9.00 

1.90 

45.00 

47.39 

3.59 

.301 

6.28 

.918 

9.50 

2.11 

50.00 

58,50 

152  PBACTICAL  HYDKAULICS: 

Ex.  76. — The  velocity  in  a  14-inch  pipe  is  4. 18  feet. 
What  is  the  total  head,  the  pipe  being  one  mile  long? 

Gal.  1st. — Find  in  velocity  column,  Table  17,  for 
"14  inches  diameters"  of  pipes,  the  given  velocity, 
4.18  feet,  opposite  which,  in  "fall  per  mile"  column,  is 
found  26.40  feet,  the  head  required  to  overcome  the 
resistances  of  the  pipe. 

In  velocity  column,  Table  18,  find  4. 17  feet,  nearest 
approximate  to  the  given  velocity,  opposite  which,   in 
"inlet  head"  column,  is  found  4.06  feet.    Then 
26.40+  .406=26.806  feet,  total  head.— Ans. 

Gal  2d.— Find,  as  by  Cal.  1st,  the  fall  26.40  feet. 

By  Rule  37: 

(4.18)2X. 02337=. 4083  feet,  .inlet  head;  26.40  + 
.4083=26.8083  feet,  total  head.— Ans. 

Ex.  77. — A  pipe,  33  inches  diameter,  being  5  miles 
long,  and  the  velocity  of  flow  in  it  9.80  feet  per  sec- 
ond, what  is  the  total  head? 

Cal.  1st.— In  Table  17,  opposite  9.80  feet,  velocity 
for  a  "33-inch"  pipe,  find  in  "fall  per  mile  column" 
47.52x5=237.6  feet,  head  due  resistances  in  pipe. 

In  Table  18,  in  velocity  column,  the  nearest  ap- 
proximate to  the  given  velocity  is  9.83  feet,  opposite 
which,  in  "inlet  head"  column,  is  found  2.257  feet; 
then  237.6+2.257=239.857  feet,  total  head.— Ans. 

Gal.  2d. — Find,  as  in  Cal.  1st,  the  friction  head= 
237. 6  feet. 

By  Rule  34: 

(9.8)2X.02337=2.244  feet,  inlet  head;  237.6  + 
2.244=239.844  feet,  total  head.— Ans. 


PKACTICAL   HYDRAULICS.  153 

Remark. — The  inlet  head,  except  in  case  of  great 
velocity,  is,  in  practice,  usually  omitted  as  insignifi- 
cant. Thus  applying  it  in  Ex.  75,  the  velocity  due 
the  friction  head,  63.36  feet,  employed  in  Cal.  4th,  is 
by  Table  17,  7.73  feet  for  an  "18-inch"  pipe. 

By  Table  18,  the  inlet  head,  due  7.74  feet  velocity 
nearest  approximate  to  7.73  feet,  is  1.40  feet,  which  is 
seen  to  be  small  in  comparison  with  the  given  head  of 
500  feet. 


EQUATIONS  AND  RULES  FOR  VELOCITY,  HEAD,  LENGTH, 
DIAMETER,  AND  VOLUME,  OF  FLOW  FOR  CLEAN  PIPES. 

The  general  equation  for  the  volume  of  flow  is: 

q=av.  (149) 

Substituting  in  (149),  the  value  of  a=  -  (area  of 
cross  section  of  pipe),  and  the  value  of  v  of  (129), 
noting  that  ^=j, 


For  convenience  of  notation  put, 


Then,  2=0  (152) 


154  PRACTICAL  HYDRAULICS. 

Transposing  (152)  successively  with  respect  to  h/,  d, 
Z,  and  reducing, 

(154, 


d=JJ^_K  (155) 

Substituting  in  Eq.  (151),  the  values  of  n=3.1416, 
2c/=64.4,  and  of  c/=. 00644,  as  a  mean  coefficient  of 
resistance  within  the  pipe: 

c,=  39.27.  (156) 

By  reference  to  Table  16,  the  coefficient  of  resist- 
ance .00644,  employed  in  finding  the  value  of  ct  of 
Eq.  (151),  is  due  a  velocity  of  2.25  feet  per  second  in 
a  6-inch  pipe,  5  feet  velocity  in  a  3- inch  pipe,  and  .7 
feet  velocity  in  a  12-inch  pipe.  Its  range  thus  appears 
too  limited  for  general  application. 

Substituting  the  value  of  cy  of  (156)  in  Eqs.  (152), 
(  153),  (154)  and  (155),  there  results: 


2=39. 27  f^5]*;  (157) 

I   *'  J 

\  d'*  \  } 
7=1542.13  j^-5j;  (159) 

£}*,-  (160) 


PRACTICAL  HYDRAULICS.  155 

Equations  (157),  (158),  (159)  and  (160),  expressed 
as  written  rules  are  as  follows: 

Rule  35. — The  quantity  of  flow  in  cubic  feet  per 
second,  iu  a  clean  pipe,  is  equal  to  39.27  times  the 
square  root  of  the  quotient  arising  from  dividing  the 
product  of  ohe  head,  and  the  5th  power  of  the  diame- 
ter, both  in  feet  measure,  by  the  length  of  the  pipe  in 
feet. 

Rule  35  corresponds  to  Eq.  (157). 

Rule  36.— The  head  is  equal  to  .000648  times  the 
quotient  arising  from  dividing  the  product  of  the 
length  of  pipe  in  feet,  and  the  square  of  the  discharge 
in  cubic  feet  per  second,  by  the  5th  power  of  the 
diameter  of  the  pipe. 

Rule  36  corresponds  to  Eq.  (158). 

Rule  37.— The  length  of  a  pipe  is  equal  1542.13 
times  the  quotient  arising  from  dividing  the  product 
of  the  head  and  the  5th  power  of  the  diameter  by  the 
square  of  the  discharge  in  cubic  feet  per  second. 

Rule  37  corresponds  to  Eq.  (157). 

Rule  38.— The  diameter  is  equal  to  .'23034  times  the 
5th  root  of  the  quotient  arising  from  dividing  the  pro- 
duct of  the  length  of  the  pipe  in  feet,  and  the  square 
of  the  discharge  in  cubic  feet  by  the  head  in  feet. 

Rule  38  corresponds  to  Eq.  (160). 

Rule  39. — The  values  of  quantity  of  flow,  the  head, 
length  of  pipe,  and  diameter  of  pipe  are  found  by 
Table  17. 


156  PKACTICAL    HYDRAULICS." 

Case  1st. — Quantity  of  flow  being  required,  di- 
vide the  given  head  by  the  given  length,  both  in  feet. 
Find  in  "sine  of  slope"  column,  the  sine  equal  to  this 
quotient,  opposite  which,  in  discharge  column  for 
the  given  diameter,  will  be  found  the  quantity  of 
flow  sought. 

Case  2d. — The  head  being  sought,  find  the  given 
discharge,  or  nearest  approximate  thereto,  in  the  dis- 
charge column  for  the  given  diameter,  opposite  which, 
in  "sine  of  slope"  column,  will  be  found  the  proper 
sine,  which,  multiply  by  the  given  length  of  pipe.  The 
product  will  be  the  head  sought. 

Case  3d. — The  length  of  pipe  being  required,  find  in 
the  discharge  column  for  the  given  diameter,  the  given 
quantity  of  flow,  opposite  which,  in  "sine  of  slope" 
column,  will  be  found  the  proper  sine.  Divide  the 
given  head  by  this  sine,  the  quotient  will  be  the  length 
of  pipe  required. 

Case  4th. — The  diameter  being  required,  divide 
the  given  head  by  the  given  length  of  pipe.  Find  in 
"sine  of  slope"  column  the  sine  equal  to  this  quotient, 
opposite  which,  in  discharge  column,  find  the  given 
quantity  of  flow,  or  nearest  approximate  thereto. 
The  diameter  for  this  discharge  will  be  the  diameter 
sought. 

Ex.  78. — The  head  being  40  feet,  the  pipe  6  inches 
diameter,  10,000  feet  long,  what  is  the  discharge  in 
cubic  feet  per  second? 

Cal.  1st. — Substituting  the  given  values  d=6"=.5 
feet,  7?/=40  feet,  and  £=10,000  feet  in  Eq.  (157); 


PBACTICAL  HYDEAULICS.  157 

^5=(.5)5X40^-10,000=.000125 1 


39.27X(.000125)i=.440  cubic  feet.—  Ana. 

Employ  Rule  39,  case  1st. 

Gal.  2d.  —  Dividing  the  given  head  by  the  given 
length  of  pipe,  s=4(H  10,000=.  004,  sine  of  slope. 

In  Table  17,  find  the  sine  of  slope,  opposite  which, 
in  discharge  column,  for  6  -inch  pipe,  is  found  the 
quantity  sought=.444  cubic  feet.  —  Ans. 

Ex.  79.  —  A  pipe  3  inches  diameter,  10,000  feet  long, 
discharges  .247  cubic  feet  per  second,  what  is  the 
head  ? 

Gal.  1st.  —  Substituting  the  given  values,  d== 
3  inches,  £=10,000  feet,  and  g=247  cubic  feet  in  Eq. 
(US)-. 


.000648X61.74=400.05  feet  head.—  Ans. 

Employ  Rule  39,  case  2. 

Gal.  2d.  —  In  Table  17,  find  in  3-in.  discharge  column, 
.2469  cubic  feet,  nearest  approximate  to  the  given 
quantity  .247,  opposite  which,  in  sine  of  slope  column, 
is  found  .04.  Then,  h=ls. 

.04X10,000=400  feet  head.—  Ana. 

Ex.  80.  —  If,  under  a  head  of  42  feet,  a  pipe  4  inches 
diameter  discharges  .3  of  a  cubic  foot  of  water  per 
second,  what  is  the  pipe's  length? 

Gal.  1st.  —  Substituting  the  given  values  of  <i=4"= 
J  foot,  &=42,  and  9=.  3  cubic  feet  in  Eq.  (159), 


158  PRACTICAL    HYDRAULICS. 

2=1542. 13 X42 X (i)5- (.3)2=2961.6  feet.— Ans. 

Employ  Rule  39,  case  3. 

Cat.  2d. — In  Table  17,  find  in  discharge  column  for 
4 -inch  pipe,  .3036  cubic  feet,  nearest  approximate  to 
the  given  quantity  .3  cubic  feet,  opposite  which,  in 

sine  of  slope  column,  is  found  .014;  then  l=-\  1= ./T2T 

s 

=3000  feet.— Ans. 

Ex.  81. — The  head  of  water  being  280  feet,  re- 
quired the  diameter  of  a  pipe  4000  feet  long,  that  will 
discharge  80,000  gallons  in  24  hours? 

Col.  1st.— q -80,000-  (21X60X60  X  7.5)  --=.124 
cubic  feet,  discharge  per  second. 

Substituting  the  given  values,  g=.l24  cubic  feet, 
7i/=280  feet,  and  2=4000  feet  in  Eq.  (160); 

d=.  23034  (4000X(.124)2-280)^=.17  feet=2.04 
inches  diameter. — Ans. 
Employ  E-ule  39,  case  4. 

Cat.  2d. — s=j=J/$\=  .01 ,  sine  of  slope. 

In  Table  17,  find  sine  of  slope  .07,  opposite  which, 
in  discharge  column,  find  .1286  cubic  feet  per  second, 
nearest  approximate  to  the  given  volume  .124.  This 
is  found  under  heading  "2  inches"  diameter.  The 
diameter  required  then  is=2  inches. — Ans. 

COEFFICIENT  OF  FLOW. 
When  the  coefficient  of  flow  in  a  long  pipe  is  c  = 


PRACTICAL   HYDRAULICS.  159 

39.27,    the  coefficient  of   velocity  deduced  from  Eq. 
(130)  or  (129),  is 

f  Ct  ,.\     1 

:100.  (161) 

In  which  case  Eq.  (130)  becomes 

i;=:100  (r  s)i  (162) 

and  Eq.  (129),  by  observing  that  f=-r, 


becomes  V=5<)  (163) 


Several  standard  authors  on  hydraulics  find  as  fol- 
lows: 

Chezy  finds  c=100.  (164) 

Eytelwein  finds  c=100.  (165) 

Leslie  finds  c=100.  (166) 

D'Aubuisson  finds  c=95.6.  (167) 

Blackwell  finds  c=95.83.  (168) 

Hawksley  finds  c=96.09  .  (169) 

Bartlett  finds  c=95.88.  (1TO) 

Jackson  finds  (small  pipes)  c=100.  (171) 

Fanning  finds  (mean  for  small  pipes), 

c=100.  (172) 

D'Arcy  finds  (for  larger  pipes)  c=113.8.  (173) 

Substituting  the  value  of  c=113.8,   as  found  by 


160  PRACTICAL   HYDRAULICS. 

D'Arcy,  the  value  of    n=3.14l6,  in  Eq.   (150),  and 
reducing, 

d*H  (174) 


COMPARISON  OF  RESULTS  OBTAINED  BY  EQUATION 
(174)  AND  BY  TABLE  17. 

If  the  head  of  water  be  100  feet,  the  pipe  1  foot 
diameter,  and  10,000  feet  long,  the  quantity  of  flow 
by  Eq.  (174),  will  be  g=4.469  cubic  feet  per  second, 
and  by  Table  17,  g=4.305  cubic  feet  per  second. 

The  head  being  16  feet,  the  pipe  4  feet  diameter, 
and  10,000  feet  long,  the  quantity  of  flow  by  Eq. 
(174)  will  be  <?=57.20  cubic  feet  per  second,  and  by 
Table  17,  g=68.50  cubic  feet  per  second. 

When  the  head  is  100  feet,  the  pipe  6  inches  diame- 
ter, and  10,000  feet  long,  the  discharge  by  Eq.  (174) 
will  be  g=.79  cubic  feet  per  second,  and  by  Table  17 
<?=  .72  cubic  feet  per  second. 

The  head  being  20  feet,  the  pipe  8  feet  diameter, 
and  10,000  feet  long,  the  quantity  of  flow  by  Eq. 
(174)  will  be  5=361.8  cubic  feet  per  second,  and  by 
Table  17  q =49 6. 4  cubic  feet  per  second. 

These  comparisons  show  that  the  coefficient,  113.8, 
proposed  as  a  mean,  is  too  large  for  small  pipes  and 
too  small  for  large  pipes;  that  it  is  adapted  to  a  pipe 
1  foot  diameter,  with  a  limited  range  for  either 
smaller  or  larger  diameters, 


PEACTICAL  HYDRAULICS' 


161 


They  illustrate,  also,  that  the  engineer,  in  the  prac- 
tice of  his  profession,  cannot  safely  venture  far  from 
an  established  fact  in  hydraulics,  without  experiment 
as  a  guide. 

BENT  PIPES. 

Bends  occurring  in  pipes  resist  the  flow  of  water 
through  them. 

To  determine  the  additional  head  requisite  to  over- 
come this  resistance,  Weisbach  (partly  on  the  experi- 
ments of  Du  Buat,  but  chiefly  on  his  own  experi- 
ments), gives  substantially  the  following  formulas  and 
tables  compiled  from  them : 

ANGULAR  BENDS. 

Let  &y=the  additional  head  required  to  overcome 
one  angular  bend. 

m=one-half  the  angle  of  deflection  of  the  bend. 
z~i}\Q  coefficient  of  bend  or  knee  resistance. 
v= velocity  of  flow.     Then, 

' ;          (175) 

0=0. 9457  sin.2  m  +  2.047  sin.*  m.  (176) 

Table  19  is  calculated  from  Eq.  (176). 

TABLE  19. 

Coefficients  for  Bend  Resistances  in  Pipe. 


m° 

z 

10° 
.046 

20° 
.139 

30° 
.364 

40° 
.740 

45° 

.984 

50° 
1.260 

55° 
1.556 

60° 
1.861 

65° 
2.158 

70° 
2.43 

162  PRACTICAL   HYDRAULICS. 

Rule  40 . — The  additional  head  required  to  overcome 
one  angular  bend  is,  in  case  the  head  generating  the 
velocity  be  given,  equal  to  the  product  of  the  given 
head,  and  the  coefficient  of  bend  or  knee  resistance, 
due  the  given  angle  of  deflection  found  in  Table  19 ; 
and  is,  in  case  the  velocity  be  given,  equal  to  the  pro- 
duct of  said  coefficient  and  the  square  of  the  given 
velocity,  divided  by  64.4. 

Ex.  82. — The  velocity  of  water  in  a  pipe,  in  which 
occurs  one  rectangular  bend,  is  10  feet  per  second. 
What  additional  head  will  be  requisite  to  overcome  the 
resistance  of  the  head? 

Gal. — By  Rule  40,  square  of    velocity  divided  by 

QAO 

64.4=10X10-64.4=1.553  feet;  m=^-=45°,  one- 
half  the  angle  of  deflection. 

By  Table  19,  the  value  of  z,  corresponding  to  45°, 
is  .984;  then  1.553X. 984=1. 528  feet,  additional 
head. — Ans. 

CURVED  BENDS. 

Let  h==  additional  head  required  to  overcome  the 
resistance  of  curvature, 

6=angle  of  curvature  of  the  pipe, 
R=radius  of  curvature  of  the  bend, 
r=radius  of  the  pipe, 
^=coefficient  of  resistance;] 


PRACTICAL   HYDRAULICS. 


163 

(178) 


Eq.  (178),  from  which  Table  20  is  computed,  is  for 
pipes   with  circular  cross  sections. 


TABLE  20. 


Coefficients  of  Resistance  of  Curvature  with  Circular  Trans- 
verse Sections. 


i  

z,  

0.1 
.131 

0.2 
.138 

0.3 

.158 

0.4 
.206 

0.5 
.294 

0.6 
.440 

0.7 
.661 

0.8 
.977 

0.9 
1.408 

1.0 
1.978 

Rule  41.— The  additional  head  required  to  overcome 
the  resistance  of  curvature  of  a  bent  pipe,  is  equal  to 

the  product  of  the  head,  &—  j  o~  r  >  generating  the  ve- 
locity, the  ratio  j  TQQO— ~  f  °f  180°  or  n>  ^°  the  an~ 

gle  of  deflection  or  length  of  bend,  and  the  coefficient 
of  resistance,  corresponding  to  the  ratio  of  the  radius 
of  curvature  to  the  radius  of  the  pipe. 

Ex.  83. — The  velocity  of  water  in  a  pipe  2  inches 
diameter,  is  16  feet  per  second:  what  additional  head 
will  be  requisite  to  overcome  the  resistances  of  a  bend 
in  the  pipe,  whose  radius  of  curvature  is  2  inches,  and 
angle  of  deflection  90°? 

Oal. — Head  generating  velocity,  equal  to  the  square 


164 


PRACTICAL  HYDBAULICS. 


of  the  velocity,  divided  by  64.4,  as  ^=16X16-^-64.4 
=3.97  5 

v  90° 

p=i=.5,  ratio  of  given  radii;  6=r-Q-Q=J,  ratio 
*•  loU 

of  180°,  to  given  angle  of  deflection. 

By  Table  20,  the  coefficient  corresponding  to  .5,  the 
ratio  of  radii  is  .294.  Then  by  Rule  41, 

3.975  X  J  X  .294=.584  feet.— Ana. 

V=0. 124+3.104  j^U.  (179) 

In  Eq.  (179),  from  which  Table  21  is  computed,  r 
represents  half  the  width  of  a  rectangular  pipe;  and 
R,  the  radius  of  curvature  of  the  axis. 


TABLE  21. 


Coefficients  of  Resistance  of  Curvature  in  Pipes  with  Rectan- 
gular Transverse  Sections. 


r 

E,"" 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0..7 

0.8 

0.9 

1.0 

z/r  .  .  . 

t.124 

.135 

.180 

.250 

.398 

.643 

1.015 

1.546 

2.271 

3.228 

Remark. — Rule  41  applies  to  finding  the  additional 
head  required  to  overcome  the  resistance  of  curvature 
of  a  bent  pipe,  whose  cross  section  is  rectangular,  by 
employing  Table  21  instead  of  Table  20. 

Total  head. — Let  in  general: 


PRACTICAL   HYDRAULICS.  165 

represent  the  additional  head  requisite  to  overcome 
the  resistances  of  n,  given  bends  —  angular  or 
curved  —  in  a  pipe,  the  form  of  whose  cross  section  is 
given  —  circular  or  rectangular. 

Combining  (180)  and  (122),  observing  that  c=.505 

of  (127),  ^=      and  2#=64.4, 


Ex.  84.-  A  pipe  1  foot  diameter,  500  feet  long,  has 
five  quadrant  or  right  angled  bends:  the  radius 
of  curvature  of  each  bend  is  1  foot;  the  velocity  of 
water  through  the  pipe  is  8.  025  feet  per  second.  What 
is  the  total  head  ? 

Gal.  By  Table  16,  the  coefficient  corresponding  to 
a  velocity  of  8  feet  per  second  in  a  pipe  whose  diame- 
ter is  d=l  ft.,  is  c/=.0052. 

Square  of  velocity   divided  by  64.4  ;  8.025X8.025 

-^-64.4=1.     In  the  present  case  n— 


r      .5 
=2.5.    Ratio  of  the  given  radii,  -=.=—=.5. 

By  Table  20,  when-^-=.5,  s=-!=.294. 

Aw  Ou 

Substituting  in  (181)  the  values  of  c,=.0052,  d= 
1  foot,  Z=500  feet,  %=2.5,  and  0a;=01=. 


feet.  —  Ans. 


166          PRACTICAL  HYDRAULICS. 


FLOW  OF  WATER  THROUGH  NOZZLES. 

The  resistance  to  the  flow  of  water  in  conically  con- 
vergent tubes,  estimated  for  the  smaller  orifice,  is  less 
than  the  resistance  in  a  cylindrical  tube  with  equal 
orifice.  Thus,  Table  13  shows  that  the  coefficient  of 
velocity  from  a  tube  converging  with  an  angle  3°  10' 
is  .894:  and  that  the  coefficient  for  a  cylindrical  tube 
of  equal  diameter  is  .829.  Let  t  represent  this  ra- 
tio: ;j' 

.     .894 

-329- 

Let  i>— the  experimental  velocity  of  water  in  a  cyl- 
indrical pipe,  the  ratio  of  whose  diameter  to  length  is 
as  I'-W')  v1=ihe  theoretical  velocity  in  same  pipe; 
cy=  coefficient  of  resistance,  varying  from  .006  to  .004; 
c/l=mean  coefficient  of  velocity  with  respect  to  the 
smaller  orifice  of  the  nozzles: 


cn=t^~  J.  (183) 

Substituting  the  values  of  <y=.004,  d=l,  andZ=10 

in  (128), 

i  (  \  i  i 

;  (184) 


V!=(20&)*,  asperEq.  (8).  (185) 

Substituting  the  values  of  v,  vlt  and  £,  in  (183), 
cn=:.836,  nozzle  coefficient  (186). 


PRACTICAL   HYDRAULICS. 


167 


TABLE    22. 

Flow  of  Water  Through  Nozzles. 

Smaller  diameter  :  to  length  :  :  1  :  10  ;  angle  of  con- 
vergence, 3°  10. 


Head. 
Feet. 

DIAMETERS    OP    NOZZLES. 

1  inch. 
Cu.  ft. 

1.5" 
Cu.  ft. 

2." 
Cu.  ft. 

2.5" 
Cu.  ft. 

3." 
Cu.  ft. 

3.5" 
Cu.  ft. 

4." 
Cu.  ft. 

4.5" 
Cu.  ft. 

10. 

.115 

.258 

.458 

.715 

1.03 

1.39 

1.92 

2.32 

12.5 

.128 

.288 

.510 

.797 

1.16 

1.56 

2.05 

2.60 

15. 

.131 

.315 

.562 

.875 

1.26 

1.72 

2.25 

2.84 

17.5 

.151 

.340 

.605 

.94 

1.36 

1.85 

2.42 

3.06 

20. 

.162 

.364 

.647 

.02 

1.46 

1.99 

2.59 

3.28 

22.5 

.171 

.386 

.686 

.08 

1.54 

2.10 

2.75 

3.48 

25. 

.182 

.407 

.725 

.13 

1.63 

2.26 

2.90 

3.67 

27.5 

.190 

.426 

.768 

.19 

1.71 

2.32 

3.03 

3.84 

30. 

.204 

.446 

.811 

.26 

1.81 

2.48 

3.24 

4.10 

32,5 

.208 

.464 

.825 

.30 

1.87 

2.53 

3.30 

4.18 

35. 

.214 

.482 

.857 

.35 

1.93 

2.63 

3.43 

4.34 

40. 

.230 

.520 

.92 

.43 

207 

2.81 

3.67 

4.64 

45. 

.242 

.553 

.97 

.52 

2.18 

2.97 

3.88 

4.93 

50. 

.256 

.583 

1.04 

1.60 

2.30 

3.13 

4.09 

5.25 

60. 

.273 

.638 

1.13 

1.77 

2.56 

3.43 

4.49 

5.75 

70. 

.307 

.690 

1.22 

1.92 

2.75 

3.75 

4.88 

6.16 

80. 

.328 

.737 

1.31 

2.04 

2.95 

4.01 

5.26 

6.64 

90. 

.347 

.778 

1.39 

2.20 

3.06 

5.54 

5.54 

7.00 

100. 

.366 

.824 

1.47 

2.29 

3.30 

5.87 

5.87 

7.41 

125. 

.410 

.92 

1.64 

2.56 

3.67 

6.75 

6.55 

8.26 

150. 

.449 

1.01 

1.80 

2.80 

4.03 

7.20 

7.20 

9.07 

175. 

.485 

1.09 

1.94 

3.02 

4.36 

7.78 

7.78 

9.80 

200. 

.518 

1.16 

2.07 

3.23 

4.59 

8.28 

8.28 

10.45 

250. 

.584 

.31 

2.32 

3.62 

5.22 

9.29 

9.29 

11.75 

300. 

.635 

.43 

2.54 

3.96 

5.67 

10.15 

10.15 

12.88 

350. 

.686 

.54 

2.74 

4.28 

6.16 

11.98 

10.98 

1385 

400. 

.733 

.65 

293 

4.58 

6.59 

11.74 

11.74 

14.82 

450. 

.778 

.75 

3.11 

4.86 

6.98 

12.46 

12.46 

15.71 

500. 

.820 

.84 

3.28 

5.12 

7.38 

13.10 

13.10 

16.60 

550. 

.859 

.89 

3.44 

5.36 

7.56 

13.75 

13.75 

17.01 

600. 

.899 

201 

3.59 

561 

8.13 

14.36 

14.36 

18.06 

700. 

.95 

2.21 

3.92 

6.11 

8.86 

15.70 

15.70 

19.93 

800. 

1.04 

2.32 

4.14 

6.47 

9.29 

16.56 

16.56 

20.90 

900. 

1.10 

2.48 

4.39 

6.87 

9.90 

17.57 

17.57 

22.27 

1000. 

1.16 

260 

4.64 

7.24 

10.40 

18.58 

18.58 

23.40 

168 


PEACTICAL   HYDRAULICS. 


TABLE  22. 

Flow  of  Water  Through  Nozzles. 

Smaller  diameter  :  to  length  :  :  1:  10  ;  angle  of  con- 
vergence 3°  10. 


Head 
Feet. 

DIAMETERS  OF  NOZZLE. 

5  inch. 
Cu.  Ft. 

5.5" 
Cu.  ft. 

6. 
Cu.  ft. 

7." 
Cu.ft. 

8." 
Cu.  ft. 

9." 
Cu.  ft 

10." 
Cu.  ft. 

12." 
Cu.  ft. 

10. 

286 

3.46 

4.13 

5.60 

7.69 

9.29 

11.44 

16.48 

125 

3.19 

3.86 

4.62 

6.26 

1.18 

10.40 

12.79 

18.44 

15. 

3.51 

4.23 

5.05 

6.86 

9.00 

11.36 

14.10 

20.20 

17.5 

3.78 

4.57 

5.44 

7.42 

9.67 

1224 

15  14 

21.75 

20. 

4.04 

4.89 

583 

7.93 

1035 

13.12 

16.18 

23.32 

22.5 

4.31 

5.17 

6.18 

8.41 

10.98 

13.92 

17.17 

24.74 

25. 

4.53 

5.47 

6.51 

9.13 

11.59 

1464 

18.08 

25.95 

27.5 

4.75 

5.74 

6.82 

9.30 

12.13 

15.35 

18.97 

27.30 

30. 

5.07 

6.14 

7.29 

9.94 

12.98 

16.39 

20,27 

29.14 

32.5 

5.15 

6.33 

7.38 

10  10 

13.20 

16.72 

20.62 

29.72 

35.0 

5.35 

6.47 

7.72 

10.49 

13.73 

17.36 

21.40 

30.86 

40. 

5.73 

6.91 

8.25 

11.24 

14.65 

18.66 

22.88 

33.00 

45. 

6.06 

7.34 

8.75 

11.89 

15.59 

19.79 

24.26 

34.98 

50. 

6.39 

7.83 

9.11 

1263 

16.35 

20.82 

25.58 

36.83 

60. 

7.00 

8.48 

10.11 

13.73 

17.92 

22.71 

28.03 

40.39 

70. 

766 

9.27 

11.02 

14.48 

19.51 

24.47 

30.27 

42.56 

80. 

8.19 

9.91 

11.81 

16.06 

21.02 

26.57 

32.36 

46.65 

90. 

8.68 

10.42 

12.46 

17.03 

22  18 

28.03 

34.75 

49.82 

100. 

9.15 

11.08 

13.18 

17.95 

23.47 

29.65 

36.63 

52.70 

125. 

10.24 

12.38 

14.69 

20.07 

26.21 

33.05 

40.96 

58.75 

150. 

11.21 

13.57 

16.13 

21.98 

28.80 

36.29 

44.86 

64.51 

175. 

12.11 

13.76 

17.42 

23.75 

31.10 

39.20 

48.47 

69.70 

200. 

12.91 

15.76 

18.58 

25.38 

33.12 

41.80 

51.80 

74.30 

250. 

15.48 

17.52 

20.88 

28.39 

37.15 

46.98 

57.92 

83.52 

300. 

15.86 

19.20 

22.90 

31.09 

40.61 

51.52 

63.45 

90.58 

350. 

17.14 

19.84 

24.57 

33.59 

4392 

5540 

68.54 

97.5 

400. 

18.31 

22.16 

26.35 

35.90 

46.94 

59.29 

73.27 

105.4 

450. 

19.43 

23.51 

27.94 

38.08 

49.82 

6?.  86 

77.72 

111.8 

500. 

20.47 

24.79 

29.52 

39.60 

52.42 

66.42 

81.92 

118.1 

550. 

21.47 

25.99 

3024 

42.10 

5501 

68.04 

85.91 

121.0 

600. 

22.44 

27.14 

32.11 

43.97 

57.46 

72.25 

89.74 

128.5 

700. 

24.46 

29.59 

35.42    47.93 

62.78 

7970 

9.79 

141.7 

800. 

25.89 

31.43 

37.15  !  50.76 

66.24 

83.52 

1036 

148.6 

900. 

26.47 

33.25 

39.60 

53.89 

70.27 

89.10 

109.8 

158.4 

1000. 

2895 

35.04 

41.62 

56.75 

74.30 

93.6 

1159 

166.5 

X 
PKACTICAL    HYDRAULICS.  169 

Ex.  85. — A  nozzle  being  6  inches  diameter  at  its 
discharge  end,  5  feet  long,  will  discharge  under  a 
head  of  150  feet  how  many  cubic  feet  of  water  per 
second? 

Col. — In  "6-inch"  diameter  column,  opposite  150 
feet  in  "head"  column,  Table  22,  the  quantity  sought 
is  found=l6.l3  cubic  feet. 


RELATIVE  CARRYING  CAPACITIES  OF  CLEAN,  FOUL 
AND  VERY  FOUL  WATER  PIPES. 

In  the  preceding  discussion  with  respect  to  the 
carrying  capacities  of  water  pipes,  they  have  been 
considered  clean. 

Assuming  the  coefficient  of  resistance  for  a  clean 
pipe,  .00644;  for  a  rough  or  foul  pipe,  .0082;  and  for 
a  very  foul  pipe,  .012,  then  will  the  relative  volumes 
of  flow  be: 

For  ctean  pipes,  1 . 0000 ; 

For  foul  pipes,  .8863; 

For  very  foul  pipes,  .7325.  : 

Rule  42. — In  case  a  water  pipe  is  rough  and  foul, 
multiply  the  flow  for  a  clean  pipe  in  Table  17  (with 
same  diameter,  length  and  head),  by  9,  8,  7,  etc.,  ac- 
cording to  the  degree  of  foulness,  as  judgment  shall 
dictate. 

Ex.  89. — A  long  pipe,  6  inches  diameter,  has  a  fall 
of  18.48  feet  per  mile.  What  will  be  the  flow  in  cubic 


170  PRACTICAL    HYDRAULICS. 

feet  per  second,  if  its  coefficient  of  discharge  with  re- 
spect to  a  clean  pipe  be  .8  ? 

Gal—  In  Table  17,  find  opposite  the  given  fall  18.48 
feet,  in  discharge  column,  for  a  6-inch  pipe,  .395  cubic 
feet.  This  is  for  a  clean  pipe  . 

Then  by  Rule  42,  .395  X.  8-=.  316  cubic  feet.—  Ans. 

Pressure  Ordinates.  —  A  mean  pressure  ordinate  is 
the  vertical  distance  between  the  axis  of  the  pipe  and 
the  hydraulic  gradient.  Thus  in  Fig.  19,  CE  is  the 
hydraulic  gradient,  and  (afc  +  r),  (cd-fr),  or  (ef+r), 
is  a  mean  pressure  ordinate.  If  the  axis  of  the  pipe 
be  depressed  below  the  base,  as  represented  in  Fig.  19, 
by  the  dotted  line,  D  a,  c,  e,  E;  a  b,  cd  or  ef,  will  be 
the  mean  pressure  ordinate  for  the  given  point,  a,  c 
or  e.  To  find  the  value  of  the  ordinate  in  pounds 
pressure: 

Let  A0=the  pressure  ordinate,  or  hight  of  water 
column  in  feet,  as  a  6;  jo—  pounds  pressure  per  square 
inch  of  the  ordinate  or  column;  whence 


fc«=-434  h0  pounds  nearly.  (187) 

Rule  43.  —  The  pressure  in  pounds  of  a  vertical  col- 
umn of  water,  whose  cross  section  is  one  square  inch, 
is  equal  to  .434  times  the  hight  of  the  column  or  or- 
dinate in  feet.  Rule  43  corresponds  to  Eq.  (187). 

THICKNESS  or  THIN  PIPES  REQUISITE  TO  WITHSTAND 

A  GIVEN  PRESSURE  IN  POUNDS  PER  SQUARE  INCH. 

Let  p=pounds  pressure  per  square  inch;  £=thiek- 


PKACTICAL    HYDRAULICS.  171 

ness  of  pipe  in  inches;  r=radius  of  pipe;  &=modulus 
of  strength,  working  load,  or  safety,  as  shall  be  given. 
By  Bartlett's  Mechanics,  page  507, 


=-.  (188) 

Rule  44.  —  The  thickness  in  inches  of  a  thin  pipe, 
requisite  to  withstand  a  given  pressure  per  square 
inch  is  equal  to  the  quotient  arising  from  dividing  the 
product  of  the  pounds  pressure  per  square  inch,  and 
the  radius  of  the  pipe  in  inches  by  the  given  modulus 
of  the  material  in  the  pipe.  Rule  44  corresponds 
to  Eq.  (188). 

Ex.  90.  —  The  water  pipe,  34.19  inches  in  diameter, 
at  the  hydraulic  works  of  the  Spring  Valley  Com- 
pany, Butte  County,  California,  laid  in  1870,  now  in 
perfect  condition,  as  reported  by  the  engineer  in 
charge,  is  subjected  to  a  working  strain  of  17,549 
pounds  per  square  inch.  The  greatest  pressure  ordi- 
nate  is  887  feet  in  hight.  What  is  the  thickness  of 
the  iron  in  the  pipe? 

Cal.—By  Rule  43,  p  =^887  X.  434=  385  pounds  pres- 
sure per  square  inch,  nearly;  r=34.  19-^-2^=  17.  095, 
radius  of  pipe. 

By  Rule  44,  £=385X  17.095-^-17,549=.  3754  inches 
thickness,  or  f  of  an  inch.  —  Ans. 

Remark.  —  The  modulus  &=17,549  pounds,  work- 
ing load,  seems  rather  too  high  for  ordinary  plate  iron. 


172 


PRACTICAL    HYDRAULICS. 


TABLE  23. 


Moduli  of  Strength,  Working  Load  and  Safety— Weisbach. 


Names  cf  Sutstances. 

Modulus  of 
Strength. 
Pounds. 

Modulus  of 
Working  load. 
Pounds. 

Modulus  of 
Safety. 
Pounds. 

Iron  iii  bars                   .  . 

59  805 

20,622 

10,311 

Iron  in  plates 

56  712 

9  280 

Cast  iron 

19  592 

14,436 

3094 

Steel  .                                   

123  700 

37,120 

20  6°2 

Copptr 

38  151 

6  187 

Lead 

329 

Box,  oak,  fir,  firm  Scotch  fir, 
pine  (American)  —  Bartlett.. 

12,373 

3,094 

1,237 

INVERTED  SIPHON. 


FIG.  20. 

A  pipe,  as  represented  by  I/y  C  G  O,  Fig.  20,  em- 
ployed in  conveying  water  across  a  valley  or  other 
depression  between  two  more  elevated  points,  as  Iy  and 
O,  is  termed  an  inverted  siphon.  It  is  a  matter  of  no 
little  importance  in  practice  to  determine  the  relation 
of  the  hights  of  inlet  and  outlet  of  an  inverted  si- 


PRACTICAL   HYDRAULICS.  173 

phon.  Thus  the  elevation,  O,  being  given,  it  is  demon- 
strable that  if  Iy/  be  the  point  of  inlet,  the  weight  of 
the  pipe  will  be  greater  than  if  the  inlet  be  taken  at 
a  lower  elevation.  It  is  also  demonstrable  if  1lu  be  the 
point  of  inlet,  the  weight  of  the  pipe  will  be  greater 
than  if  the  inlet  were  taken  at  a  higher  elevation. 

For  example,  having  given  the  elevations  of  the 
several  stations,  and  distances  between  them,  of  an 
inverted  siphon,  whose  length  is  I  feet,  let  it  be  re- 
quired to  find  the  hight  of  the  inlet  point,  above  the 
outlet  point,  so  that  the  weight,  "W,  pounds  of  the 
material  employed  in  the  construction  of  the  siphon, 
carrying  q,  cubic  feet  of  water  per  second,  shall  be  a 
minimum? 

Let  1= length  of  siphon  in  feet; 

W~ weight  of  siphon  in  pounds; 

g= cubic  feet  flow  of  water  per  second; 

2r 

d=^  diameter  of  siphon  in  feet; 

t=  thickness  of  shell  of  siphon  in  inches; 

w= weight  of  a  cubic  foot  of  material  in  siphon; 

h— working  modulus  of  strength  of  material; 

j9=mean  pressure  in  pounds  per  square  inch; 

x— hydraulic  head,  such  that  the  weight,  W,  shall 
be  a  minimum. 

Let,  in  Fig.  20,  O  represent  the  point  of  outlet; 
O  C  (approximately  equal  to  O  CT),  a  horizontal  line, 
intersecting  the  pipe  line  in  C;  O  A=a,  determined 
from  the  given  levelling  data,  the  ordinate  of  mean 


174  PBACTIOAL    HYDKAULICS. 

hydrostatic  pressure,  in  the  part,  O  G  C,  of  the  siphon 
filled  to  the  level,  O  C;  then 


} 


U 


Let  the  perpendicular,  C  I—x,  so  that  the  area  of 
the  triangle,  O  C  I,  shall  equal  the  area  of  O  G  Iy. 

Then  will  the  mean  pressure  ordinate  in  feet  for 
the  entire  pipe,  be: 


(190) 


In  most  cases,  without  considerable  error,  C I  may 
be  taken  equal  to  C,  Iy,  thereby  assuming  that  the 
area,  C  P  Iy,  is  equal  to.  the  area,  O  P  I. 

If  greater  accuracy  be  required,  make  O  A  of  such 
a  hight  that  the  trapezoid,  G  A  By  It,  shall  equal  the 
area  embraced  by  the  pipe  line,  Iy  C  G  O,  and  the  hy- 
draulic gradient,  OIy.  In  case  the  exact  length  of 
the  pipe  is  not  given,  but  the  approximate  horizontal 
distance,  OCy=ABy,  is  known  approximately,  put 

ABy-l — ^-^r =^,  length  of  pipe. 

Then 


Substituting  the  value  of  E  F=&0  in  (187), 


PEACTICAL  HYDRAULICS.  175 

Substituting  the  values  of  p  of  (191),  and  of  r= 

in  (188), 

t_1.808d 


1C 

The  obvious  equation  for  the  weight  of  the  siphon 

W=fi</-';.  (193) 


Substituting  the  value  of  t  of  (192)  in  (193), 


Squaring  both    members  of    Eq.  (155),    observing 
that  x=hf> 


Substituting  the  .value  of  df  of  (195)  in  (194), 
.  1085  n  w  $ 


c*k  \    xt    ) 

Differentiating  (196),  omitting  the  constant  factor 
outside  the  parenthesis, 

^=_^-i  +  |.-f=0.  (197) 

doc  o 

An 

Reducing  (197),  x=-f.  (198) 

Decomposing  (197)  into  factors,  and  differentiating 


176  PRACTICAL    HYDRAULICS. 

that  factor  which  reduces  to  0  (zero),  the  second  dif 
ferential  becomes: 

d2W     4a- 


which,  being  positive,  determines  that  "W  is  a  mini- 
mum, when  x=-^,  as  found  in  Eq.  (198). 

o 

SPECIAL  VERIFICATION. 

To  verify,  by  special  example,  the  correctness  of 
the  result,  showing  that  the  weight,  W,  of  an  in- 
verted siphon,  carrying  a  given  quantity,  q,  of  water, 

is  a  minimum,  when  the  head,  hf=x=-^-:     substitute 

o 
the  value  of  #==-— ,  also  different  values,  one  greater 

o 

and  one  less  than  -~-,  as  x=-n-,  and  #  =— -  in  Eq. 

o  O  o 

(196). 

For  convenience  of  notation,  let  /;=the  constant 
factor  outside  of  the  parenthesis.  The  substitutions 
being  made: 

~~3' 

W=2.971/,a*;  (200) 

\\r-i.  Z>a 

When  #,=-«-, 

W/=2.989//  or;  (201) 


PRACTICAL  HYDRAULICS.  177 

Sa 
And  when  x4t~-n-—a, 

Wy/=3/>l     *          (202) 

Since  the  literal  factors  in  Eqs.  (200),  (201)  and 
(202)  are  identical,  the  numerical  factors  in  these 
equations  express  the  relative  weights  of  W,  Wy  and 
W/;  in  the  given  order. 

These  factors  conclusively  show  that  the  greatest 
economy  is  attained  in  the  weight  of  the  siphon,  when 

4a 

the  value  of  x—-~-t  as  determined  in  Eq.  (198). 
o 


MINIMUM  WITH  RESPECT  TO  PRESSURE  ;  DIAMETER, 
THICKKESS,  AND  WEIGHT  OF  AN  INVERTED  SIPHON— 
MINIMUM  MEAN  PRESSURE  PER  SQUARE  INCH. 


Substituting  the  value  of  OS=-Q-  in  Eq.  (191), 


(203) 
MINIMUM  DIAMETER. 


a 
Substituting  in  (155),  the  values  of  x=-     of  (198), 

=hf, 
(204) 


c,=3.l514-T  Y  of  (151)'  and  recollecting  that  ii=hf, 


178 


•  MINIMUM  MEAN  THICKNESS. 

Substituting  the  values  of  fi=-q-,  and  of  d,  of  (2041) 
in  (192), 

(205) 


MINIMUM  WEIGHT. 


Substituting  the  values  of  dt  of  (204),  and  of  t  of 
(205)  in  (193), 

(206) 

In  obtaining  the  value  of  (206),  the  inner  diameter 
of  the  pipe  has  been  employed — the  same  as  used  with 
respect  to  discharge — whereas  the  shell  of  the  pipe 
being  exterior  to  this  diameter,  its  weight  is  somewhat 
greater  than  represented  by  the  value  of  W  (206),  as 
will  readily  appear  by  the  following: 

Let  A=area  of  cross  section  of  pipe  with  respect  to 
internal  diameter. 

A7=area  of  cross  section  with  respect  to  external 
diameter. 

Then  A=5rf,';  (207) 


PEACTICAL    HYDRAULICS.  179 

And  A--d+2t2==d'+^dt  +        .  (208) 


Deducting  (207)  from  (208),  and  decomposing  the 
difference  into  factors, 

A—  A=ftd,*l+-.  (209) 


Farther,  the  pipe  has  been  regarded  seamless, 
whereas,  in  most  cases,  water  pipes  —  especially  the 
larger  class  —  are  constructed  with  rivets  and  laps,  or 
bands,  whose  weight  must  be  added  to  that  of  W  of 
(206). 

Let  n=  this  weight,  expressed  in  form  of  percent- 

age; thus: 

W/-W  (l+n).  (210) 

Let  W;/=the  entire  weight  of  pipe,  including  cor- 
rections shown  by  (209)  and  (210). 

Then  as  fid,  t  only,  as  respects  area  of  cross  section 
of  shell,  is  involved  in  the  value  of  W  of  (206),  will 

(211) 


If  the  pipe  is  very  thin,  the  factor  -1  1  +  -?  r  may  be 
omitted;  if  seamless  (l-\-n),  will  be  omitted. 

TO  FIND  THE  MEAN  HYDRAULIC  PRESSURE  BELOW  THE 
LEVEL  OF  THE  OUTLET. 

Rule  45.  —  Find  in  square  feet,  from  the  levelling 


180  PRACTICAL   HYDRAULICS. 

data,  the  area  embraced,  as  represented  in  Fig.  20, 
within  the  boundary  of  the  line  of  pipe,  O  G  C  (axis 
of  pipe),  and  the  horizontal  line,  O  C,  drawn  from  the 
outlet,  O,  and  intersecting  the  pipe  line  in  C.  Divide 
this  area  by  the  length  of  the  line,  O  C,  in  feet,  the 
quotient  will  be  the  mean  hydrostatic  ordinate  in  feet, 
represented  by  OA.  The  length  of  this  ordinate, 
multiplied  by  the  decimal,  .484,  will  be  the  mean 
hydrostatic  pressure  in  pounds  sought. 
Rule  45  corresponds  to  Eq.  (189). 


TO  FIND  THE  HYDRAULIC  HEAD,  SUCH  THAT  THE 
WEIGHT  OF  THE  MATERIAL  EMPLOYED  IN  THE  CON- 
STRUCTION OF  THE  PIPE  CARRYING  A  GIVEN  QUANTITY 
OF  WATER,  SHALL  BE  A  MINIMUM. 


Rule  46. — Divide  four  times  the  length  of  the  hy- 
drostatic ordinate,  as  found  according  to  Rule  45,  by  3; 
the  quotient  will  be  the  hydraulic  head,  CI  (Fig.  20), 
required. 

Rule  46  corresponds  to  Eq.  (198). 

Remark. — In  Fig.  20,  I  represents  the  point  at 
which  the  pressure  in  the  pipe  begins.  This  point  de- 
termined, that  of  I;  will  readily  be  found,  from  the 
levelling  data,  and  will  require  to  have  an  additional 
head  sufficient  only  to  discharge  the  given  quantity  of 
water  at  I. 


PRACTICAL    HYDRAULICS.  181 


TO  FIND  THE  MEAN  ORDINATE,  E  F,  FlG.  20,  IN- 
VOLVING BOTH  THE  HYDRAULIC  HEAD,  0  I,  AND  THE 
HYDROSTATIC  ORDINATE, 'A  O. 


47, — Divide  the  sum  of  the   hydraulic  head, 
C  I,  and  twice  the  hydrostatic  ordinate  by  2. 
Rule  47  corresponds  to  Eq.  (190). 


TO  FIND  THE   MEAN   PRESSURE   PER   SQUARE   INCH   IN 
POUNDS  FOR  THE  ENTIRE  PIPE. 


Rule  48. — Multiply  the  mean  hydrostatic  ordinate, 
O  A,  Fig.  20,  in  feet  by  the  decimal  .7234. 
Rule  48  corresponds  to  Eq.  (203). 


To  FIND  THE  MINIMUM  DIAMETER. 


Rule  49. — The  minimum  diameter  is  equal  to  .5965 
times  the  fifth  root  of  the  quotient  arising  from  divid- 
ing the  product  of  the  coefficient  of  resistance,  the 
length  of  the  pipe  and  the  square  of  the  discharge  per 
second,  by  the  hight  of  the  hydrostatic  ordinate,  O  A, 
Fig.  20,  in  feet. 

Rule  49  corresponds  to  Eq.  (204). 


182  PRACTICAL  HYDRAULICS. 


TO  FIND  THE  MINIMUM  MEAN  THICKNESS. 


Rule  50. — Multiply  the  fifth  root  of  the  product  of 
the  coefficient  of  resistance,  the  length  of  the  pipe, 
the  square  of  the  discharge  per  second,  and  the  fourth 
powfjr  of  the  mean  hydrostatic  ordinate,  O  A,  Fig.  20, 
by  the  quotient  arising  from  dividing  2.5908  by  the 
modulus  of  the  working  load  or  of  safety,  as  shall  be 
required. 

Rule  50  corresponds  to  Eq.  (205). 


TO  FIND  THE  MINIMUM  WEIGHT. 


Rule  51. — Case  1. — The  pipe  being  very  thin,  mul- 
tiply the  fifth  root  of  the  product  of  the  square  of 
the  coefficient  of  resistance,  the  seventh  power  of  the 
length  of  the  pipe,  the  fourth  power  of  the  discharge 
per  second,  and  the  cube  of  the  mean  hydrostatic  or- 
dinate, 0  A,  Fig.  20,  by  the  quotient  arising  from 
dividing  .4046  times  the  weight  of  a  cubic  foot  of  the 
material  in  the  pipe  by  the  modulus  of  the  material. 

Case  2. — The  pipe  being  thick  as  y\  of  an  inch  or 
more,  and  seamless,  multiply  the  result  obtained,  ac- 
cording to  Case  1,  by  1  (unit),  increased  by  the  quo- 
tient arising  from  dividing  the  thickness  of  the  shell 
by  the  inner  diameter. 


PRACTICAL    HYDRAULICS.  188 

Case  3. — The  pipe  being  thick,  as  T3^  of  an  inch  or 
more,  and  constructed  with  rivets  and  laps  or  bands, 
multiply  the  result  obtained,  according  to  Case  2,  by 
1  (unit),  increased  by  their  relative  weight  to  that  of 
the  pipe. 

Rule  51  corresponds  to  Eq.  (211). 

The  moduli  with  respect  to  the  strength,  working 
load,  and  safety  are  given  in  Table  23. 

The  modulus  of  working  load,  as  shown  in  Ex.  90, 
is  &=17,549  pounds. 

Unless  the  iron  is  extra  in  quality,  the  modulus 
ought  to  be  less,  as  &=14,000. 

The  weight  of  a  cubic  foot  of  iron  is  usually  esti- 
mated at  480  pounds. 


184 


TEACTICAL    HYDRAULICS. 


TABLE  24. 


Number,   Thickness    and    Weight    of   One    Square    Foot  of 
Sheet  Iron. 


BIRMINGHAM  GAUGE. 

AMERICAN  GAUGB.—  HASWELL 

No. 

Thi'k 
in. 

Lbs. 

No 

Thi'k 
in. 

Lbs. 

No. 

Thi'k 
in. 

Lbs. 

No. 

Thick 
in. 

Lbs. 

0000 

.454 

18.35 

17 

.058 

2.34 

0000 

46 

18.63 

19 

.036 

1.45 

000 

.425 

17.18 

18 

.049 

1.98 

000 

.41 

16.58 

20 

.032 

1.29 

00 

.38 

15.36 

19 

.042 

1.70 

00 

.365 

14.77 

21 

.028 

1.15 

0 

.34 

13.74 

20 

.035 

1.42 

0 

.325 

13.15 

22 

.025 

1.03 

1 

.3 

12.13 

21 

.032 

1.29 

1 

.289 

11.70 

23 

.023 

.913 

r 

.284 

11.48 

22 

.028 

1.13 

2 

.258 

10.43 

24 

.020 

.814 

c 

t! 

.259 

10.47 

23 

.025 

1.01  i 

3  .229 

9.29 

25 

.018 

.724 

4 

.238 

9.62 

24 

.022 

.889 

^ 

.204 

8.27 

26 

.016 

.644 

£ 

.22 

8.89 

25 

.02 

.808 

5 

.182 

7.37 

27 

.014 

.574 

e 

.203 

8.21 

26 

.018 

.723 

6 

.162 

6.56 

28 

.013 

.511 

7 

.18 

7.28 

27 

.016 

.647 

7 

.144 

5.84 

29 

.011 

.455 

8 

.165 

6.67 

28 

.014 

.566 

8 

.128 

5.20 

30 

.010 

.405 

9 

.148 

5.98 

29 

.013 

.525 

9 

.114 

4.63 

31 

.009 

.360 

10 

.134 

5.42 

30 

.012 

.485 

10 

.102 

4.13 

32 

.008 

.321 

11 

.12 

4.85 

31 

.010 

.404 

11 

.091 

3.67 

33 

.007 

.286 

12 

.109 

4.41 

32 

.009 

.364! 

12 

.081 

3.27 

34 

.0063 

.254 

13 

.095 

3.84 

33 

.008 

.323 

13 

.072 

2.92 

35 

.0056 

.226 

14 

.083 

3.36 

34 

.007 

.283 

14 

.064 

2.59 

36 

.005 

.202 

15 

.072 

2.91 

35 

.005 

.202 

15 

.057 

2.31 

37 

.0045 

.180 

16 

.065 

2.63 

36 

.004 

.162 

16 

.051 

2.05 

38 

.004 

.159 

17 

.045 

1.33 

39 

.0035 

.142 

18 

.040 

1.63 

40 

.0031 

.127 

Ex.  91. — The  following  data  from  a  sheet-iron  in- 
verted siphon  being  given,  viz: 
Length  of  pipe,  128.5  miles. 

^Elevations  with  respect  to  sea  level: 
<(      Point  of  inlet,  1300  feet. 
I     Point  of  outlet,  350  feet. 


PRACTICAL    HYDRAULICS.  185 

Mean  hydrostatic  ordinate,  as  O  A,  Fig.  20,  305.5 
feet;  discharge  of  water  per  second,  37.57  cubic  feet; 
modulus  of  safety  of  the  iron,  14,000  pounds;  weight 
of  iron  per  cubic  foot,  485  pounds;  allowance  for  bands, 
laps  and  rivets,  15  per  cent;  cost  of  laid  pipe  per 
pound,  10  cents.  Required,  the  minimum  diameter, 
thickness  of  shell,  weight  and  cost  of  the  siphon  ? 
Required,  also,  the  diameter,  thickness  of  shell,  weight 
and  cost  of  the  siphon,  if  950  feet,  the  full  hydraulic 
head,  be  employed  ? 

Cal.  1st. — The  given  hydrostatic  ordinate  is  305.5 
feet. 

By  Rule  46,  corresponding  to  Eq.  (198),  305.5X4-^- 
3=407.34  hydraulic  head;  407.34-f-128.5=3.17  feet 
fall  per  mile. 

By  Table  17,  it  is  seen  that  for  3.17  feet  fall  per 
mile,  the  pipe  carrying  37.57  cubic  feet  per  second 
will  approximate  48  inches  in  diameter,  and  that  the 
corresponding  velocity  is  3.20  feet  per  second. 

By  Table  16,  for  a  velocity  oi  3  feet  in  a  48-inch 
pipe,  the  coefficient  of  resistance  is=.0038. 

By  Rule  49,  corresponding  to  Eq.  (204), 

. 5965  (•^H2<i^|-y<^jj<l3_7,ii)2j15-  =  3898  feet  = 
46.776  inches,  minimum  diameter  (internal). — Ans. 

By  Rule  50,  corresponding  to  Eq.  (205), 

{2f||«|.(.0038xl28.5X5280x(305.5)4}*=.3694 
inches  in  thickness. — Ans. 

By  Rule  51,  corresponding  to  Eq.  (211), 

oV-  { (-0038) 2  (128.5  X 5280)7(37. 57)4(305. 5)3  p 


186  PEACTICAL  HYDBAULICS. 

.15  =  143790750     pounds,     minimum 


weight.  —  Ans. 

Whence  at  lOc.  per  pound:  Cost=$14,379,075.00.— 
Ans. 

CW.  2d.—  1300—  350=950  feet  total  head;  950-^- 
128.5-=7.392  feet  fall  per  mile. 

By  Table  17,  the  diameter  of  pipe  having  7.392  feet 
fall  per  mile,  and  discharge  37.57  cubic  feet  of 
water  per  second  is—  40  inches. 

By  Rule  47,  corresponding  to  Eq.  (190),  (305.  5  X 
2+  950)+-  2=780.5  feet  mean  ordinate  for  the  entire 
pipe. 

By  Rule  43,  corresponding  to  Eq.  (187),  780.5  X 
.434=338.737  mean  ordinate  in  pounds  for  the  entire 
pipe. 

By  Rule  44,  corresponding  to  Eq.  (188),  338.737 
X20-^  14000=.  4839  inches  thickness  of  pipe;  40X 
3.1416-^-12=10.472  feet  circumference  of  pipe; 
10.472  X  128.  5  X5280X  485  X.  4839-*-  12  =  138954840 
pounds  weight  of  pipe,  assumed  seamless,  and  esti- 
mated for  the  internal  diameter. 

By  Rule  51,  corresponding  to  Eq.  (211),  cases  2  and 
3,  138954840(1  +  -^^)  XL  15  =  161,747,880  pounds 
weight  of  pipe,  employing  the  full  head  of  950 
feet.  —  Ans. 

Whence,  at  lOc.  per  pound:  Cost-$16,174,788.00. 

—  Ans. 

The  difference  in  these  results,  viz.,  $16,174,788.00 

—  $14,379,075.  00=$1,795,713.00,  which  amounts  to 


PRACTICAL  HYDRAULICS'     *  -  187 

a  saving  of  over  11  per  cent  by  the  application  of  the 
principle  hereinbefore  demonstrated  with  respect  to 
the  minimum  weight  and  minimum  cost  of  an  in- 
verted siphon. 

There  will,  in  fact,  be  a  greater  saving  in  practice, 
arising  from  a  less  length  of  pipe  under  pressure,  in 
case  of  employing  the  smaller  head. 


FLOW  OF   WATER   IN   OPEN   CHANNELS  AND   NATURAL 
STREAMS. 


The  flow  of  water  in  open  channels  and  natural 
streams  is  subject  to  the  same  laws  which  govern  its 
flow  in  pipes.  The  force  producing  motion  in  the 
water,  and  overcoming  the  resistances  of  the  water 
way,  is  that  of  gravity  applied  to  an  inclined  plane. 
A  greater  variety  of  forms,  with  respect  to  cross  sec- 
tion of  streams,  is  presented  in  open  channels  and 
natural  streams  than  in  pipes,  thereby  changing  to  a 
greater  extent  the  relations  between  the  perimeters 
and  the  areas  of  the  cross  sections  of  the  former,  than 
of  the  latter.  Thus,  in  the  case  of  pipes,  the  "hy- 
draulic mean  radius"  has  been  shown,  uniformly,  equal 
to  one-fourth  of  the  diameter,  while  in  open  channels 
their  mean  depths  vary  indefinitely. 

The  "mean  depth"  of  an  open  channel  or  natural 
stream  is  the  ratio  of  the  perimeter  to  the  area  of  the 
cross  section  of  the  stream.  For  the  most  part  in  hy- 


188  *      PEACTICAL  HYDRAULICS. 

draulic  computations,  that  portion  of  the  perimeter 
which  bounds  the  bottom  and  sides  of  this  area, 
termed  the  "wet  perimeter,"  is  employed. 

The  "air  perimeter,"  whose  value  does  not  often  ex- 
ceed one-tenth  of  an  equal  length  of  the  wet  perime- 
ter, unless  strong  winds  or  other  disturbing  influences 
obtain,  is  considered  when  great  accuracy  is  required : 

Let  a  =. area  of  cross  section  of  stream; 

£>=_wet  perimeter; 

m p—Siir  perimeter; 

m= coefficient  of  air  perimeter; 

T= hydraulic  mean  depth. 

Thenr=--  (212) 

P 

(213) 


p-f  mp 

Other  things  being  equal,  the  greater  the  ratio  of 
the  perimeter  to  the  area  of  the  cross  section  of  a 
stream  of  water,  the  less  will  be  the  resistance  to  flow. 

The  forms  of  cross  sections,  generally  applied  to 
water  ways,  are  rectangular  and  trapezoidal. 

FORM  OF  RECTANGLE  OF  MAXIMUM  CARRYING  CAPA- 
CITY. 

Let  p= perimeter  (omitting  air  perimeter) ; 

o;=hight  of  rectangle; 

Then  p — 2#— width  of  rectangle. 


PRACTICAL  HYDRAULICS.  189 

— — 2~=r,  maximum.  (21 4) 
P 

Differentiating  (214),  x=£,  hight;  (215) 

$^3a>±M  width.  (216) 


FORM  OF  TRAPEZOID  OF  REGULAR  FIGURE  OF  MAX- 
IMUM CARRYING  CAPACITY. 


In  Fig.  21,  let  £=BAE,  angle  of  slope  of  bank; 
g=a  side;  then  hight=f  sin  t.  Mean  width— f  -j-| 
cos  -t: 

^2(sin  £  +  sin  t  cos  t)—r  maximum.  (2>7) 

Differentiating  (217),  observing  that  sin2  £=1 — cos2  2, 
cosM  +  ^-W  (218) 

Reducing  (218),  cos  *=i=.5.  (219) 

By  table  natural  sines,  £=60°.  (220) 

Of  the  regular  figures,  the  semi-circle  consisting  of 
an  infinite  number  of  sides,  so  that  at  any  point  cos  t 
=l=r,  offers  the  least  resistance  to  flow. 

By  equations  (215)  and  (216),  it  is  seen  that  the 
form  of  a  rectangle,  offering  the  least  resistance  to 
flow,  has  its  base  or  width  equal  to  twice  its  hight; 
and  by  equations  (219)  and  (220),  it  is  seen  that  of  the 


190  PRACTICAL   HYDRAULICS. 

regular  figures,  the  trapezoid  whose  angle  of  slope  is 
60°,  in  other  words  the  semi-hexagon,  offers  the  least 
resistance  to  flow. 

THE  ANGLE  OF  SLOPE  AND  THE  AREA  BEING  GIVEN  TO 
DETERMINE  THE  MOST   APPROPRIATE  FORM  OF  A  CANAL. 


B 

iie.  21. 


Let  in  Fig.  21,  p=A. B  +  B  C  +  CD—  perimeter;  b= 
BC=bottom;  Wangle  A;  n  =  cot  t;  a— area;  x—d= 
B  E,  depth  of  canal.  Then 

ix-  (221) 

')i;  (222) 

2)  ;  (223) 

a=(b+nx)x-,  (224) 


-  .  ,^^^r^ 

whence  b=~  (225) 


x 


Substituting  value  of  b  of  (225)  in  (223)  and  divid- 
ing then  both  members  by  a, 


minimum.  (226) 

X        CL  ^ 

Differentiating  (226)  and  reducing, 

(227) 


PRACTICAL    HYDRAULICS . 


Substituting"    the     values   of    TI— cot  £—  - 


COS  t 

sin  t 


" 


sin* 


—  cos 


From  (225)  b=——x  cot  £. 


191 

and 

(228) 
(229) 


TABLE  25. 


Dimensions  of  the  most  suitable  forms   of  Canals,   corres- 
ponding to  different  angles  of  slopes,  and  to  a  given 
area  of  cross  section. 


Angle  of 
Slope  =t. 

Ratio  of 
Perp.  to 
Base. 

Relative 
Slope. 
n. 

Depth. 
d' 

v« 

Bottom 
Width. 
b 

ft 

nd 

ft 

Top 
Width. 

b+2nd 

T/a 

Perimeter. 
P 

l/« 

90°  00' 

lonO 

.0 

0.707 

1.414 

.0 

1.414 

2.828 

78°  41' 

Son  1 

0.200 

0.734 

1.217 

0.147 

.510 

2.713 

75°  58' 

4onl 

0.250 

0.734 

1.161 

0.186 

.533 

2.692 

71°  34' 

3onl 

0.333 

0.752 

1.079 

0.251 

.580 

2.656 

03°  26' 

2onl 

0.500 

0.759 

0.938 

0.379 

.697 

2.635 

60°  00' 

26  on  15 

0.577 

0.760 

0.877 

0.439 

.755 

2.632 

56°  19' 

3  on  2 

0.667 

0.759 

0.812 

0.506 

.824 

2635 

53°  8' 

4  on  3 

0.750 

0.757 

0.753 

0.568 

1.892 

2.645 

51°  20' 

5  on  4 

0.800 

0.753 

0.724 

0.603 

1.960 

2.654 

45°  00' 

lonl 

1.000 

0.740 

0.613 

0.740 

2.092 

2.704 

40°  00' 

21  on  25 

1.192 

0.722 

0.525 

0.860 

2.246 

2.771 

36°  52' 

3  on  4 

1.333 

0.707 

0.471 

0.943 

2.557 

2.828 

35°  00' 

7  on  10 

1.402 

0.697 

0.439 

0.995 

2.430 

2.870 

33°  41' 

?on3 

1.500 

0689 

0.418 

1.034 

2.465 

2.989 

30°  00' 

23  on  40 

1.732 

0.664 

0.356 

1.150 

2.656 

3012 

26°  34' 

Ion  2 

2.000 

0.636 

0.300 

1.272 

2.844 

3.144 

21°  48' 

2on5 

2.500 

0.589 

0.228 

1.471 

3.170 

3.397 

18°  26' 

Ion  3 

3.000 

0.548 

0.188 

1.645 

3.478 

3.646 

14°  2' 

Ion4 

4.000 

0.485 

0.119 

1.941 

4.001 

4.121 

11°  19' 

Ion  5 

5.000 

0.441 

0.062 

2.205 

4.472 

4.519 

semi-cir. 

0.798 

1.596 

2.507 

192  PEACTICAL    HYDRAULICS. 


TO  FIND  THE  DIMENSIONS  OF  THE  MOST  SUITABLE 
FOEM  OF  CANAL,  WHEN  THE  ANGLE  OF  SLOPE  OF  THE 
BANKS  AND  THE  AEEA  OF  CEOSS  SECTION  AEE  GIVEN. 


Rule  52.  —  Employing  Table  25,  multiply  the  square 
root  of  the  given  area  of  cross  section  by  the  num- 
ber in  the  table,  which  is  opposite  the  given  angle  of 
slope  of  bank,  and  in  the  column  of  the  denomination 
of  the  dimension  sought,  as  "depth,"  "bottom  width," 
"top  width,"  "perimeter"  (wet). 

Ex.  92.  —  What  dimensions  must  be  given  to  the 
cross  section  of  a  canal  —  trapezoid  of  regular  form  — 
whose  discharge  of  64  cubic  feet,  with  a  velocity  of  4 
feet  per  second,  is  a  maximum? 

Gal—  By  Eq.  (220),  it  is  shown  that  the  angle  of 
slope—  60°,  when  the  carrying  capacity  of  a  regular 
trapezoidal  canal  is  a  maximum. 

64-=-  4  =16  square  feet,  area  of  cross  section. 

(16)^  =  4  square  root  of  area  of  cross  section. 
By  Table  25,  Rule  52,  opposite  60°,  the  angle  of 

slope  in  "depth"  column,  find  .760;  that  is,  -^--=.760; 


or 

But,  as  shown  above,  the  square  root  of  the  area  o? 
cross  section  is=4  feet;  hence,  d,  =  .760X4  =  3.  01 
feet,  depth  of  canal.  —  Ans. 

Farther,  opposite  60°,  in  "bottom   width"  column, 


PRACTICAL  HYDRAULICS.  193 

find  .877;  that  is, -7-- =.877;   or  6=.877i/t«,  but 'as 

1/66 

shown  above  ]/c6=4;  hence,  &=. 877X4=3.508  feet, 
bottom  width. — Ans. 

Farther,  opposite  60°  find  in  "  top  width  col- 
umn" 1.755  feet;  that  is, b  +  2nd===  1.755  feet;  or  b  + 

"]/  (A> 

_2?id=l. 755X4=7.020  feet,  top  with.—  Ans. 

Farther,  opposite  60°  in  "perimeter  column," 
find  2.632;  that  is,  p=2. 632X4=10.528  feet  peri- 
meter.— Ans. 

Ex.  93. — The  following  data  for  a  canal  in  loose 
earth  being  given,  viz  :  Discharge  of  water,  50  cubic 
feet  per  second;  velocity  of  flow,  2  feet  per  second ; 
angle  of  slope,  26°  34'  equivalent  to  1-2;  it  is  re- 
quired to  determine  the  dimensions  of  the  most  appro- 
priate form  of  cross  section  of  the  canal  ? 

Cal. — 50-^-2=25  square  feet,  area  of  cross  section; 
l/ ~ =1/25=5  square  root  of  area  of  cross  section. 

By  Table  25,  opposite  26°  34',  the  given  angle  of 
slope  (l'-2),  in  depth  column,  find  .636;  that  is 

4=  .636;  or  c/y=.636  i/a. 

I  a 

Substituting  value  of  j/a=5;  in  this,  d/=. 636x5= 
3.18  feet,  depth.— Ans. 

Farther,  in  column  of  ''bottom  width,"  find  6= 
.300  i/a,  or  by  substituting -5  for  /«,  Z>=. 300X5=1.5 
feet,  bottom  width. — Ans. 

Farther,  in  column  of  "top  width,"  find  />X  2  H  <l( 


194  PRACTICAL  HYDRAULICS. 

2.844  i/o,  or  by    substituting  5   for   *\/'a,  b  -}-2nd= 
2.844X5=14.22  feet,  top  width.— Ans. 


P. 

p 

/;=3.144X5=15.72  feet,  perimeter.— Ans. 


Farther,  in  "perimeter"  column,  find  4==3.144,  oi- 
l/a 


FORMULAS    FOR  THE  FLOW    OF  WATER    IN   OPEN 

STREAMS. 


Fquation  (130)  is  an  expression  for  uniform  velocity 
of  water  in  open  streams,  as  well  as  in  pipes.  In  case 
the  air  perimeter  of  an  open  stream  be  considered,  the 

value  of  r  .=  —        -  of  (213)  must  be  substituted  for 


r  in  (130);  but  in  case  the  air  perimeter  be  omitted, 
equation  (130)  for  the  velocity  of  water  in  pipes  re- 
mains unchanged  for  the  velocity  of  water  in  open 
streams,  that  is: 


Iii    the    factor  -  —  i-2'  c/  is  a  variable  coefficient, 

(  cf  ) 

whose  value  depends  upon  experiment.  This  simple 
equation  covers  all.  cases  in  practice,  providing  the 
value  of  Cf  be  known. 

Of  the  numerous  formulas  in  use  for  finding  the 
velocity  of  water  in  open  streams,  the  following  from 
Kutter,  taken  in  connection  with  his  table  of  coeffi- 


PRACTICAL  HYDRAULICS.  195 

cients  for  roughness  of  stream  bed,  seems  entitled  to 
the  foremost  position,  on  account  of  its  directness  and 
wide  range  of  application. 

KUTTER'S  FORMULA. 


In  (231),  v,  T  and  s,  respectively,  denote  velocity, 
hydraulic  mean  depth,  and  sine  of  slope,  and  n  repre- 
sents the  coefficient  of  roughness  of  the  stream  bed. 

Eq.  (231)  may  be  somewhat  simplified  by  putting 


;  (232) 


When 


The  first  factor  of  the  second  member  of  Eq. 
(231)  is  equal  to  the  first  factor  of  the  second  mem- 
ber of  Eq.  (230),  and  being  deduced  from  a  wide 
range  of  experiments  with  great  care  and  masterly 
skill,  furnishes  the  best  means  known  to  the  present 
writer  of  determining,  in  the  absence  of  actual  expe- 

riment in  a  special  case,  the  value  of  -j  Let  ck 

=this  expression  from  Kutter's  formula;  then 

v=ck(rs)*.  (234) 


196  riUCTICAL    HYDRAULICS. 


TABLE    26. 

Coefficients  (n)  for  Roughness  of  Stream    Beds.     Compiled 
from  Kutter,  Jackson  and  Fanning. 

72,=  .009  well  planed  timber  in  perfect  order. 

n— .010  cement,  glazed,  coated  material. 

%  =  .01 2  unplaned  timber  in  flumes. 

7i=i.0l3  brickwork,  cast  and  wrought-iron. 

7i  =  .015  canvas  lining,  rectangular  flumes  with  bat- 
tens .5  inch  apart. 

?i  =  .017  rubble,  also,  earth  in  highly  regular  cases. 

7i=.020  coarse  rubble,  set  dry,  in  bad  condition, 
very  firm  regular  gravel,  and  flumes  with  batcens  2 
inches  apart. 

n— .0225,  dry,  coarse  rubble  in  bad  order;  earth 
canals  and  channels  above  average. 

7i— .0250  earth,  canals  and  channels  in  good  order. 

7i— .0275  earth,  canals  and  channels  below  average. 

7i=.030  earth,  canals  and  channels  in  bad  order. 

7i=.035  rivers  and  canals  in  bad  order,  overgrown 
with  vegetation,  and  strewn  with  stones. 

-^=.070  rivers  in  earth,  with  stones  and  weeds  in 
great  quantities. 

The  principal  elements,  besides  the  force  of  gravity 
involved  in  the  determination  of  the  velocity  of  an 
open  stream  of  water,  are  the  hydraulic  mean  depth, 
the  sine  of  slope,  and  the  degree  of  roughness  of .  the 
stream  bed.  Now,  if  these  relations  be  not  changed, 
it  is  evident  that  the  velocity  of  the  water  will  neither 


PRACTICAL   HYDRAULICS.  197 

be  increased  nor  diminished  by  varying  the  form  and 
magnitude  of  the  sectional  area  of  the  stream. 

Illustrative  of  this  proposition: 

1st. — The  hydraulic  mean  radius  or  mean  depth  of 
a  circular  pipe,  in  which  d=diameter,  is: 


p     4tnd     4* 

2d. — The  hydraulic  mean  depth  of  a  V  flume  (uvee 
flume"),  right  angled  at  the  bottom,  in  which  <i=the 
slant  hight  or  width  of  side, 

a         d*         d  f9W\ 

r=p=2X2d^  (236) 


3d.     The  hydraulic  mean  depth  of    a  rectangular 
me  or 

depth  is, 


flume  or  canal,   in  which  <i=the  width,  and  ~= 


d*         d  QP7X 

="  (237) 


4th.     To  determine  a   trapezoidal  flume   or  canal, 

in   which   the   mean   width   and    the  angle  of  slope 

of  the  sides  are  given,  such  that  the  hydraulic  mean 

depth  shall  be  equal  to  one  fourth  of  the  mean  width. 

Let  demean  width. 

x=  width  of  side  or  slant  hight. 

2/=bottom  width. 

t  —angle  of  slope  of  sides.' 


198  PRACTICAL    HYDRAULICS. 

Then 


x  sin  2=  depth. 

(338) 

xcost  +  y=d. 

(239) 

dx  sin  £     d 

(240) 

2x  +  y       4  ' 

Whence:  a 

d 

(241) 

cos  t  -f-  4  sin  £  —  2  ' 

Anrl  -?/  —  rl  J 

4  sin  t—  2       ) 

^24^ 

Case  1st.— Put  £=45°. 

By  Table   of  natural  sines  and  cosines,  sin  45°  = 
.70711;  cos  45°=.70711. 

Substituting  values  of  sin  t  and  cos  t  in  (241)  and 
(242), 

aj=.6512  d.  (243) 

2/=.5395d.  (244) 

Depth—a  sin  £=.4605  d.  (245) 

a     .4605  cT=d 


Case  2nd.— Put  *==60°. 

By   Table  of  natural   sines  and -cosines,  sin  609-— 
.86603;  cos  60°=. 50000. 

Substituting  values  of  sin  60°  and  cos  60°  in  (241) 
and  (-242), 

^=.5091  d.  (247) 

2/=.7454  d.  (248) 

Depth  x  sin  £=.4409  cl.  (249) 

p       .4409cf    d 

And  ^-        (250)- 


PRACTICAL   HYDRAULICS.  199 

By  giving  different  values  to  t  in  (241)  and  (242), 
the  form  of  sectional  area  of  a  trapezoidal  flume  or 
canal  may  be  varied  indefinitely  without  change  of 

the  hydraulic  mean  depth,  ^=T* 

The  forms  given  in  which  £=45°  and  £=60°,  seem 
to  occur  most  frequently  in  practice. 

Now,  as  under  like  conditions,  excepting  the  forms 
of  stream  beds,  the  velocity  of  water  in  flumes  or 
canals  is  equal  to  its  velocity  in  pipes  of  equal  hy- 
draulic mean  radii  or  depths.  Table  17  giving  the 
velocity  of  flow  in  pipes,  applies  equally  well  to  this 
class  of  flumes  or  canals. 


TO  FIND  THE  VELOCITY  OF  WATER  IN  A  FLUME  OR 
CANAL  WHOSE  HYDRAULIC  MEAN  DEPTH  IS  EQUAL  TO 
THAT  OF  A  PIPE  OF  GIVEN  DIAMETER. 


Rule  53. — Case  1st — The  flume  being  V  shaped, 
that  is,  quadrant  in  form  (usually  termed  "vee  flume)," 
find  opposite  the  given  sine  of  slope  or  fall  per  mile, 
in  Table  17,  the  velocity  of  flow  due  a  pipe  whose 
diameter  is  equal  to  the  side,  width  or  slant  hight  of 
the  flume.  The  quantity  so  found  will  be  the  ve- 
locity sought.  « 

Case  2d. — The  flume  or  canal  being  either  rectangu- 
lar or  trapezoidal,  find,  opposite  the  given  sine  of  slope 
or  fall  per  mile  in  Table  17,  the  velocity  of  flow  due 


200  PRACTICAL  HYDRAULICS. 

a  pipe  whose  diameter  is  equal  to  the  mean  width  of 
the  flume  or  canal.  The  quantity  so  found  will  be 
the  velocity  sought. 


TO  FIND  THE  FLOW  OF  WATER  IN  CUBIC  FEET  IN  A 
FLUME  OR  CANAL  WHOSE  HYDRAULIC  MEAN  DEPTH  IS 
EQUAL  TO  THAT  OF  A  PIPE  OF  GIVEN  DIAMETER. 


Rule  54. — Case  1st. — The  flume  being  V  shaped, 
quadrant  in  form,  a  "  vee  flume,"  find  the  velocity  as 
directed  in  Rule  53,  arid  multiply  the  velocity  so  found 
by  one  half  the  square  of  the  side  width  or  slant- 
depth  of  the  ratio. 

Case  2nd. — The  flume  or  canal  being  rectangular, 
multiply  the  velocity  found  as  directed  in  Rule  53,  by 
one  half  the  square  of  the  width  of  the  water  way. 

Case  3d. — The  flume  or  canal  being  trapezoidal,  if 
the  angle  of  the  slope  of  the  sides  is  equal  to  45°,  mul- 
tiply the  velocity  found  as  directed  in  Rule  53,  by 
.4605  times  the  square  of  the  mean  width;  if  the 
angle  of  slope  of  the' sides  is  equal  to  60°,  multiply 
the  velocity  by  .4409  times  the  square  of  the  mean 
width  of  the  water  way,  and,  in  general,  if  the  angle 
of  slope  of  the  sides  be  equal  to  t,  multiply  the  veloc- 
ity found  as  directed  in  Rule  53,  by  the  product  of 
the  square  of  the  mean  width,  and  the  ratio  of  the 
mean  width  to  the  vertical  depth  of  the  water  in  the 
flume  or  canal. 


PRACTICAL  HYDRAULICS.  201 

Rule  55. — Another  method  of  finding  the  flow  of  a 
given  flume  is  to  multiply  the  discharge  of  a  circular 
pipe,  Table  17,  of  equal  fall  per  mile,  and  equal  hy- 
draulic mean  depth,  by  the  ratio  of  the  sectional  area 
of  the  pipe  to  the  sectional  area  of  the  given  flume; 
that  is,  if  the  flume  is  either  quadrant  in  form,  a  "V 
flume,"  or  if  it  is  rectangular,  having  its  width  equal 
to  twice  its  depth,  multiply  by  .637;  if  it  is  trape- 
zoidal, multiply  by  .  586  when  the  angle  of  slope  of  the 
side  is  45°,  and  by  .561  when  it  is  60°. 


COEFFICIENT  OF  ROUGHNESS  INVOLVED  IN -TABLE  17. 


The  measure  of  the  degree  of  roughness  of  the  inner 
surfaces  of  the  pipes  for  which  the  flow  of  water  un- 
der pressure  has  been  computed  in  Table  17,  closely 
approximates  .011  when  referred  to  Kutter's  scale  of 
coefficients  (Table  26  of  the  present  work) :  that  is 

71=^.011. 

Ex.  94.— The  fall  per  mile  being  15.84  feet,  what 
will  be  the  velocity  and  discharge  per  second  of  water 
in  a  "  V  flume,"  whose  side  width  or  slant  depth  of 
water  is  2.5  feet,  and  the  coefficient  of  roughness  of 
inner  surfaces  equal  to  .011  ? 

Cat. — With  respect  to  velocity  by  Rule  53,  in 
Table  17,  opposite  the  given  fall  15.84,  in  velocity 
column  of  2.5  feet  diameter,  find  5.27  feet. — Ans. 


202  PRACTICAL    HYDRAULICS. 

•Calculation  with  respect  to  discharge:  One-half  the 
square  of  side,  2.5X2.5-^2=3.125  square  feet. 

By  Rule  51,  5.27X3.125=16.47  cubic  feet.— 4ns. 

Or,  by  Rule  55,  find,  by  Table  17,  the  discharge 
of  a  pipe  2.5  diameter,  for  the  given  fall,  25.87  cubic 
feet. 

25.87X-637=16.48  cubic  teet.—Ans. 

Ex.  95.— The  fall  being  5.28  feet  per  mile,  what 
will  be  the  discharge  of  water  flowing  in  a  flume 
whose  mean  width  is  8  feet,  angle  of  side  slope  45°, 
and  coefficient  of  roughness,  w=.011. 

Cat.— By  Table  17,  for  a  fall  of  5.28  feet  per  mile, 
and  8-foot  pipe,  the  discharge  per  second  is  350.5 
cubic  feet.  Then  by  Rule  55, 

350.5  X- 586=205. 39  cubic  feet.— Ans. 

Remark. — With  respect  to  the  flow  of  water  in 
pipes  not  differing  largely  in  size,  it  may  be  assumed 
without  material  error  in  practice,  that  the  velocities 
are  proportionate  to  the  respective  diameters. 

If  greater  accuracy  be  required,  recourse  to  (133) 
will  need  be  had . 

Ex.  96. — The  fall  being  2.64  feet  per  mile,  what  is 
the  discharge  per  second  in  a  flume  9  feet  wide,  4.5 
feet  deep,  and  the  coefficient  of  roughness,  w=.011? 

Gal—  By  Table  17,  the  velocities  for  a  fall  of  2.64 
feet  per  mile,  due  pipes  8  and  10  feet  diameter — 


PKACTICtL  HYDKAULICS-  203 

=^1-5-=9   feet— are    4.88,    and    5.84    feet— 
=^-^-J-^-^=5.36  feet,  velocity  per  second. 

Whence,  5.36x9x9-5-2= 217.08  cubic  feet.  —Ans. 


K  utter's  Formula  for  the  Flow  of  Water  in 
in  Open  Streams. 


Among  the  most  eminent  experimeiitists  in  hy- 
draulics, during  the  present  half  century,  have  been 
D'Arcy  and  Bazin,  Humphreys  and  Abbot,  and  Kut- 
ter.  Prior  to  this  time,  the  science  of  hydraulics  was 
largely  speculative  and  incoherent.  The  older  hy- 
draulicians  had  determined  on  meager  data  certain 
laws  which  they  erroneously  held  susceptible  of  gen 
eral  application. 

D'Arcy  and  Bazin  having  collected  a  large  amount 
of  experimental  data,  deduced  therefrom  and  published 
in  1835  and  1865,  a  formula  better  adapted  than  any 
preceding  for  finding  the  flow  of  water  in  open  streams 
and  pipes  of  medium  size.  The  report  of  Humphreys 
and  Abbot  on  the  '  'Physics  and  Hydraulics"  of  the  Mis- 
sissippi river,  published  in  1861,  is  very  justly  esteemed 
by  engineers,  first  in  importance,  as  to  the  extent,  ac- 
curacy, and  value  of  its  contributions  to  experimental 
hydraulics.  Their  formulas,  however,  deduced  from 
their  experiments,  besides  being  quite  complex  and 


204  PRACTICAL   HYDRAULICS. 

tedious  of  application,  give  results  too  low  for  the  flow 
of  water  in  small  and  medium  sized  streams.  Thus, 
between  the  formulas  of  D'Arcy  and  Bazin,  and  those 
of  Humphreys  and  Abbot,  there  existed  a  wide  hiatus, 
till  it  was  effectually  closed  up  in  1870  by  the  intro- 
duction of  Kutter's  formula. 

The  mode  in  biief  of  Herr  Kutter,  in  accomplishing 
this  difficult  and  laborious  task,  is  substantially  as 
follows: 

1st.; — To  divide  the  great  mass  of  the  observed  and 
trustworthy  results  at  his  command,  appertaining  to 
the  flow  of  water  in  open  streams,  into  twelve  classes, 
arranged  as  shown  in  Table  26,  with  reference  to  the 
degree  of  roughness  of  the  stream-beds. 

2d.— To  adopt,  on  careful  comparison  of  various 
formulas  for  the  velocity  of  water  under  the  imposed 
conditions,  the  following  formula  of  Chezy  as  the 
basis : 

v=c(rs)z,  (251) 

noting  that  c  is  variable. 

3d. — For  the  determination  of  the  values  of  c,  that 
shall,  when  substituted  with  the  given  values  of  r;  the 
hydraulic  mean  depth,  and  of  s,  the  sine  of  slope  in 
the  Chezy  formula  (251),  yield  results  respectively 
corresponding  to  those  determined  by  observation,  he 
makes  extensive  experiments  with  several  trial  formu- 
las, among  which  is  that  of  D'Arcy  and  Bazin,  hitherto 
considered  in  our  discussion  of  the  flow  of  water  in 
pipes. 


PRACTICAL   HYDRAULICS.  205 

Of  these  trial  formulas,  the  following  finally 
adopted  is  one  devised  by  himself,  which  is  not  only 
more  simple  in  form  than  that  of  D'Arcy  and  Bazin, 
but  yields  results  nearer  in  accord  with  those  observed: 

a 
c= V.  (252) 

This  formula,  however,  is  faulty,  in  that  it  is  limited 
in  application. 

Thus  by  inversion,  it  is  shown  to  be  an  equation  t<5 

a  straight  line  whose  abscissa=— ^-,  whereas  the  plot- 
ted results  of  observation  indicate  a  curve,  and  further 
show  that  of  is  dependent  upon  the  value  of  b" — in  a 
word,  that  a'  and  V  are  variables,  and  not  constants, 
as  at  first  assumed. 

4th. — To  generalize  Eq.  (252)  the  variable  terms 
z  for  ci  and  x  for  6'  are  substituted  in  it;  whence, 


(253) 


5th.  —  "After   much  examination  and  further  com- 
parison," (Kutter's  words)  he  puts, 

z=a  +  ^-;  (254) 

x—n  z  —  l=a  n.  (255) 


206  PRACTICAL    HYDRAULICS. 

Substituting  these  values  of  z  and  x  in  (253),  we 
have 


(256) 


Equation  (256)  is  found,  however,  not  suited  to  the 
extremes  of  inclination  of  water  surface,  nor  to  the 
extreme  limits  of  sectional  area. 

«  6th.  —  To  meet  these  requirements,  in  other  words 
to  render  the  formula  applicable  to  all  cases  whatever, 
Kutter  noting  that  "when  r=  infinity,  c  wi\l=z,  and 
the  coefficients  z  will  have  their  values  represented  by 
an  hyperbolic  curve,"  makes  in  Eq.  (253), 

z=&  +  ~,  (257) 

s 

in  which  A  denotes  the  semi-axis  of  an  hyperbola,  m 
the  tangent  of  the  inclination  of  its  asym  totes  with 

the  axis  of  abscissa,  and  —  (s  representing  the  sine  of 
slope)  the  abscissae;  whence, 


(259) 


Substituting  the  values  of  z  and  x  of  Eqs.  (258)  and 
(259)  in  (253),  there  results  the  equation  in  its  general 
form  for  the  value  of  the  coefficient  c,  viz: 


PRACTICAL  HYDRAULICS.  207 


£        m 

n       s  /n^AA 

.  c=—  (260) 


Substituting  the  value  of  c  of  Eq.  (260)  in  (251), 
there  results  the  general  formula  of  Kutter  for  the  ve- 
locity of  water  in  open  streams,  viz: 


(261) 
1   I 
4] 

Combining  (257)  and  (254), 


Making  s  infinite  in  Eq.  (262), 


A=a+-.  (263) 


To  determine  the  relation  of  c  to  71,  in  its  simplest 
form,  let 

1=1,  (264) 

in  which  £—1,  as  found  by  trial. 

Substituting  this  value  of  I,  and  of  -7  in  Eq.  (256), 
and  reducing,  we  obtain  the  relation  sought, 

-=n:  (265) 

c 


208  PRACTICAL  HYDRAULICS. 

which,  as  determined  for  the  Mississippi  river,  is 

w=-~.027.  (266) 

c 

The  value  -of  A  in  an  hyperbola,  coinciding  with  the 
curve  formed  by  plotting  observed  results,  is  found 
to  be: 

A=60.  (267) 

Substituting  the  values  of  A=60  of  (267),  1=1  of 
(264),  and  ^=.027  of  (266)  in  (263),  transposing  and 
reducing, 

a=A— -=60— 37=23.  (268) 

To  find  the  value  of  tangent  m,  let  an  extreme  case 
be  taken,  in  which  the  values,  as  determined  from  the 
plotted  curve,  are: 

8=0.00000363;  (269) 

0=487.  (270) 

Substituting  these  values,  together  with  that  of  A== 
60  of  (267)  in  (257),  transposing  and  reducing, 

m=(z— A)s=(487-60)X0.00000363=0.00155.    (271) 

Subsituting  these  constant  values  of  a=23  of  (268), 
2=1  of  (264),  and  w=0.00155  in  (261),  there  results 
for  metrical  measures: 


«H-  nnniV,  ::>M*-  (272) 


PRACTICAL    HYDRAULICS.  209 

To  reduce  metrical  measures  employed  in  Eq.  (272) 
to  those  of  a  different  system,  let  e  denote  the  ratio  of 
the  former  to  the  latter,  noting  that  n  and  s,  repre- 
senting ratios,  are  not  affected  by  the  reduction. 


Substitute  ?=23  +  --h>;  (273) 

.  Ti  S 


(274) 


(275) 

Let  z,  x',  r  and  v  represent  respectively  the  terms 
to  which  z,x,  r  and  v,  of  the  metrical  system  are  to  be 

z  x'  r        ,        v' 

reduced,  then  will  z— — ,  x~— ,  r=~  and  v=-.  - 
e  e  e  e 

Substituting  these  values  of  z}  x,  r  and  v  in  (275) 

(276) 


Multiplying  in  (276),  both  numerator  and  denomi- 
nator within  the  parenthesis  by  e4;  also,  multiplying 
both  sides  of  the  equation  by  e, 


210  PRACTICAL  HYDRAULICS. 

z  x' 

Substituting  the  values  of  ~=e^z,  and  -f—e^tx.  in 

e-2  62 

(277). 


With  respect  to  the  Kutter  formula,  equation  (278) 
is  general  for  the  reduction  of  the  measures  employed 
in  it  to  those  of  another  system.  An  inspection  shows 
that  by  multiplying  z  and  x,  by  e*  (the  square  root  of 
the  ratio  of  the  different  measures),  each  term  of 
the  equation  [1  (unit)  being  common,]  will  be  the  de- 
nomination sought. 

To  render  the  equation  in  terms  of  English  feet, 

Let  e=3.281,  the  number  of  feet  in  a  meter.       (279) 

Substituting  the  value  of  e*=1.811  in  (278)  after 
restoring  the  values  of  z  and  x  of  (273)  and  (274), 
and  omitting  the  accents  with  respect  to  v  and  r', 


Equation  (280)  for  the  purposes  of  application  may 
be  somewhat  simplified  in  the  following  manner. 

Putting  the  numerator  inclosed  in  brackets  under 
the  following  form  : 


PRACTICAL   HYDRAULICS.  211 


Numerator^-      41.6  +    '  +  1-811>        (281) 


and  putting  x=^l.6+n.  (282) 

S  x 

Substituting  x  for  its  value  in  (280),  and  then 
multiplying  both  numerator  and  denominator  by  r% 
there  results: 


=_  .          s  I.  (283) 

n 


APPLICATION  OF  THE  KUTTER  FORMULA  AS  REN- 
DERED BY  EQS.  (282)  AND  (283). 


Ex.  97. — In  a  rectangular  flume  or  canal  four  feet 
wide,  two  feet  deep,  the  fall  per  mile  is  4.752  feet 
(equivalent  to  a  sine  of  slope,  s=.0009),  and  the  co- 
efficient of  roughness  of  whose  bed  is  7i=.025.  What 
is  the  velocity  of  flow  per  second? 

Gal. — Area  of  cross  section,  4X2=8;  wet  peri- 
meter, 4+2+2—8. 

Hydraulic  mean  depth,  8-^-8=1. 

Square  root  hydraulic  mean  depth.   ri=/i=l. 

Square  root  of  sine  of  slope,  si=i/.^=.03. 

Substituting  the  given  values  of  s=.0009,  and  n— 
.025  in  Eq.  (282), 

. 025=1418. 


212  PRACTICAL   HYDRAULICS. 

Substituting  the  value  of  ?i=.025,  and  the  values 
as  found  of  ^=1.118,  s*=.03,  and  r*=l  in  Eq.  (283), 

L.118  +  1.81K 
1.118  +  1    )X-°8' 


Reducing,  ^=4orx  .03=1.658  feet.—  Ans. 


Ex.  98.  —  In  a  rectangular  flume  or  canal  four  feet 
wide,  two  feet  deep,  the  fall  per  mile  is  4.752  (equiva- 
lent to  a  sine  of  slope  s  —  .0009),  and  the  coefficient  of 
roughness,  of  whose  bed  is  n—  .012,  what  is  the  ve- 
locity of  flow  per  second? 

Cat.—  It  will  be  noted  that  Ex.  98  differs  from  Ex. 
97,  as  relates  to  coefficient  of  roughness  of  the  bed 
only. 

Substituting  the  given  value  of  ?i=.012,  the  values 
as  found  of  £=.536,  s*=.03,  and  r*=l  in  Eq.  (283), 


Reducing,  v=83.33    j  X  .03  =  3.82  feet.— Ans. 

Ex.  99.- — It  is  required  to  construct  a  rectangular 
canal,  whose  fall  per  mile  shall  be  2.112  feet  (equiva- 
lent to  sine  s=.0004),  and  coefficient  of  roughness  of 
bed  71=. 025,  what  must  be  its  depth  and  width,  for 
it  to  discharge  108.2  cubic  feet  per  second? 

Gal. — Let  q— a  v  =  108.2  cubic  feet,  the  discharge 
per  second.  '  (a) 


PRACTICAL   HYDRAULICS.  213 

Substitute  the  value  of  v  of  (283)  in  Eq.  (a), 

arxK  +  l.Sllx  .  M 

q-=  —  (  —  L_  —  —  \s*.  (b) 

n\    x~t~i*    ) 

Arranging  terms  of  Eq.  (6)  with  respect  to  r, 

*  (c\ 


Substituting  the  values  of  s  and  n  in  (282), 


In   a   rectangular   canal,   whose   depth   is   d,    and 
width  2d,  the  hydraulic  mean  depth  is: 

d 


area,  a=2d2.  (/) 

Substituting  the  values  of  a?—  1.216,  s^  —  .02,  ^= 

025,  r=|,  a=2cZ2,  and  5=108.2  in  Eq.  (c),  and  ob- 

erving  that  d*=d'£t 

cZ*—  31.59  di=54.33. 


214  PRACTICAL    HYDRAULICS. 

Solving  Eq.  (g)  by  Hutton's  method  for  the  resolu- 
tion of  equations  of  the  higher  order: 


—31.59 

54.33 

2 

4 

8 

16 

32. 

.82 

9 

g 

24 

64 

.41 

53.51 

4 

12 

32 

80 

160. 

43.06 

2 

12 

48 

160 

54.89 



10  4"> 

6 

24 

80 

240 

215.3 

8.65 

2 

16 

80 

34.5 

62.3 



1QA 

8 

40 

160 

274.5 

277.6 

1.79 

2 

20 

12.5 

37.1 

10.7 



10 

60 

172.5 

311.6 

288.3 

2 

2.4 

13. 

39.8 

10.8 

12 

62.4 

185.5 

351.4 

299.1 

2.5 

13.5 

6.4 

64.9 

199.0 

357.8 

2.5 

14. 

67.4 

213. 

2.6 

2. 

70.0 

215. 

2.6 

2.236 


72.6 

Thus  #=2.236.  (h) 

Squaring,  d= 5  feet  depth;  whence  2d  =  10  feet 
width. — Ans. 

Ex.  100. — In  a  rectangular  canal  two  feet  deep, 
four  feet  wide,  the  fall  per  mile  is  4.752  feet,  equiva- 
lent to  sine  of  slope  s=.0009,  and  the  observed  ve- 


PRACTICAL    HYDRAULICS.  215 

locity  per  second  2.578  feet,  what  is  the  value  of  the 
coefficient  n  for  the  roughness  of  the  bed? 

Gal—  Let  m=.41.6+^?i.  (a) 

s 

Substituting  m  of  (a)  for  its  value  in  (282), 

x=mn.  (b) 

Substituting   mn   for  x  in  (283),    and   arranging 
terms  with  respect  to  n, 

/•v  r*  —  r  shn\ 
n*-f(-  -lnp=1.8llr>3  (c) 

V          V  771          / 

Substituting  the  value  of  s=.0009  in  Eq.  (a), 

44.7.  (d) 


From  the  given  data,  the  hydraulic  mean  depth  is 

found  to  be: 

r=4X2-*-(4-f2  +  2)=l.  (e) 

Substituting  in  Eq.  (c),  the  values  of  ??i=44.7  of 
Eq.  (d),  r=l,  r-=l,  ss=.03,  and  the  given  value  of 

v=2.578  feet. 

7i2+.  01072^=.  000471.  (/) 

Completing,  square  ^2+J1+(.00536)2=,.  0004997.   (^J 

Extracting  root   and   transposing,  n  =.017.  —  Ans. 

Ex.  101.  —  In  a  rectangular  canal  two  feet  deep, 
four  feet  wide,  the  coefficient  of  roughness  of  the  bed 
is7i=.012,  the  observed  velocity  of  flow  4.946  feet 
per  second,  what  is  the  sine  of  slope? 


216 


PRACTICAL  HYDRAULICS. 


Cal. — Arranging  terms  of  Eq.  (280),  with  respect 
to  s, 

s      /41.6  nz  v4-nv  rK   2     /-        .00281?" 


r-f-1 


7it;rK  |     /        .  00 
.811r/88  +  Ul.6  w  r 


__ 
+  l.SlTr 


(") 


Substituting  the  given  values  of  t;— 4.946,  n= 
.012,  and  the  value  of  .r=l,  r'  — 1,  as  found  in  the 
preceding  example,  in  Eq.  (a): 

si— .03852  si  -f-  .00001459  s*=.  0000008663 .  (I) 

Solving  Eq.  (b)  by  Hutton's  method  for  the  resolu- 
tion of  cubic  equations: 

.03872 


-  .03852 
.03 

+  .0000146 
-  .0002556 

+  .0000008663 
-  .000007230 

-  .00852 
.03 

-.0002410 
.0006444 

.C000080963 
.OU00070336 

.02148 
.03 

.0004034 
.0004758 

.0000010647 
.0000010306 

.05148 
.008 

.0008792 
.0005398 

.0000000341 
.0000000308 

.05948 
.008 

.0014190 
.0000533 

33 

.06748 
.008 

.0014723 
.0000538  • 

.07548 
.0007 

.0015261 
.0000015 

.07618 
.0007 

.0015276 

.07688 
.0007 

.07758 


PRACTICAL   HYDEAULICS.  217 

Thus  8*=.  03872.  (c) 

Squaring,  s=.00l5,  sine  of  slope. — Ans. 

Remark. — It  will  be  observed  that  Eq.  (c)  obtained 
in  the  solution  of  Ex.  98,  Eq.  (c)  in  that  of  Ex.  99, 
and  Eq.  (a)  in  that  of  Ex.  (100),  are  general  and  ap- 
plicable for  the  solution  respectively  of  all  similar 
problems. 

Table  27  has  been  computed  by  the  author  of  the 
present  work  direct  from  Eqs.  (282)  and  (283),  for  the 
velocity  of  water  in  open  streams  differing  in  regime 
and  slope,  and  varying  from  the  size  of  a  small  ditch 
to  that  of  the  Mississippi  river.  The  computation  has 
been  made  for  four  different  values  of  n,  to-wit:  n= 
.012,  ?i=.0l7,  n=  .025,  and  w=.035. 

(For  explanation  of  the  values  of  n,  see  Table  26.) 

Thus,  the  hydraulic  mean  depth  being  r=-~  =1,  and 

the  fall  per  mile  F=5.28  feet,  or  sine  of  slope  s=.001, 
the  velocity  per  second  for  the  respective  values  of  n, 
as  shown  by  Table  27,  are: 

When  %=.012,  the  velocity  is  v= 4.028  feet; 
7i=.0l7,  the  velocity  is  v=2.720  feet; 
71,=. 025,  the  velocity  is  ?;— 1.755  feet; 
7i=. 035,  the  velocity  is  #=1.190  feet. 

In  the  several  headings  of  the  table,  F  represents  in 
feet  the  fall  per  mile,  and  s  the  equivalent  sine  of 
slope. 

Table  28  has  been  prepared  to  be  used  in  connection 


218  PEACTICAL  HYDRAULICS. 

with  Table  27.  In  the  trapezoidal  forms  of  canal 
beete,  considered  -in  this  table,  the  bottom  and  sides  are 
equal  each  to  each  in  the  same  cross  section. 

Table  29  has  been  computed  to  facilitate  in  finding 
the  wet  perimeter  of  the  bed  of  a  trapezoidal  canal. 

Let  6=bottom  width; 

d—  depth  of  water  —depth  of  bank; 

m=ratio  of  depth  to  base  of  bank; 

m  d=base  of  bank; 

y=slope  of  bank; 

jp=wet  perimeter. 


Then  y=d  (1  +  m2)*;  (284) 

p=2d  (1  +  m2)  *  +  &.  (285) 


The  computed  values  of  the  base  and  bank  slope 
for  a  unit  depth  are  ari&nged  under  their  respective 
headings. 


PRACTICAL   HYDRAULICS. 


219 


TABLE  27. 


Flow  of  Water  per  Second  in  Open  Streams,  the  Coefficient 
of  Roughness  of  whose  beds  is  n=.O12. 


if    s  *• 

^"l 

^">T 

w^J  |f 

21? 

—  "    II  || 

^|y 

^»?^ 

^IT 

J^I 

!•!•§§ 

g.  2.  §  g 

|£|i 

lllg 

go*  of" 

I  ||E 

i-2-§® 

•a?  2 

•IS|H 

•3~$ 

VJ   ai** 

•  *.2j» 

•4S§ 

•4fp.S 

<PF£ 

•vrp^ 

•  frsa 

.25 
.3 

.0856 
.1016 

.2627 
.3064 

.4126 
.4780 

.6250 
.7204 

.7864 
.9046^ 

.9206 
1.058 

1.038 
1.192 

1.144 
1.313 

.4 

.1329 

.3894 

.6004 

.8989 

1.134 

1.312 

1.477 

1.612 

.5 

.1635 

.4676 

.7146 

1.062 

1.325 

I.t45 

1.739 

1.912 

.6 

.1925 

.5416 

.8213 

1.214 

1.513 

1.762 

1.981 

2.178 

.7 

.2226 

.6123 

.9227 

1.359 

1.690 

1.966 

2.209 

2.428 

.8 

.2508 

.6804 

1.020 

1.495 

1.856 

2.159 

2.425 

2.664 

.9 

.2796 

.7*57 

.111 

1.625 

2.016 

2.343 

2.631 

2.889 

1. 
1.26 

.3073 
.3752 

.8090 
.9493 

.201 
.411 

1.761 

2.044 

2.170 
2.527 

2.520 
2.932 

2.828 
3.287 

3.034 
3.608 

1.5 

.4408 

1.100 

.606 

2.316 

2.858 

3.312 

3.711 

4.071 

2. 

.5665 

1.359 

.960 

2.806 

3.453 

3.995 

4.474 

4.905 

2.5 

.6366 

1.596 

2.281 

3.247 

3.987 

4.606 

5.158 

5.652 

3. 

.8017 

1.815 

2.576 

3.651 

4.477 

5.171 

5.783 

6.335 

3.5 

.9127 

2.021 

2.852 

4.121 

4.930 

5.691 

6.362 

6.967 

4. 

.9968 

2.216 

3.111 

4.380 

5.356 

6.189 

6.905 

7.561 

4.5 

.124 

2.402 

3.357 

4.714 

5.759 

6.640 

7.419 

8.122 

5. 

.226 

2.579 

3.590 

5.030 

6.141 

7.077 

7.822 

8.653 

5.5 

.324 

2.741 

3.814 

5.333 

6.506 

7.495 

8.370 

9.160 

6. 

.421 

2.977 

4.029 

5.624 

6.856 

7.896 

8.615 

9.647 

6.5 

.516 

3.135 

4.236 

5.903 

7.192 

8.281 

9.244 

10.11 

7. 

.608 

3.290 

4.435 

6.171 

7.515 

8.651 

9.655 

10.56 

7.5 

.699 

3.443 

4.629 

6.433 

7.831 

9.012 

10.06 

11.00 

8. 

.783 

3.590 

4.817 

6.686 

8.134 

9.359 

10.44 

11.42 

8.5 

.875 

3.656 

4.999 

6.931 

8.430 

9.695 

10.81 

11.83 

9. 

.961 

3.792 

5    TJB 

7.169 

8.715 

10.02 

11.18 

12.26 

9.5 

.046 

3.926 

5  3TO 

7.402 

8.996 

10.34 

11.54 

12.61 

10. 

2.128 

4.056 

5!518 

7.632 

9.268 

10.66 

11.88 

12.99 

11. 

2.292 

4.309 

5.846 

8.067 

9.798 

H.25 

12.55 

13.73 

12. 

2.450 

4.551 

6.157 

8.485 

10.29 

11.83 

13.19 

14.42 

13. 

2.604 

4.785 

6.458 

8.889 

10.78 

12.39 

13.81 

15.09 

14. 

2.754 

5.010 

6.748 

9.273 

11.25 

12.92 

14.40 

15.74 

15. 

2.900 

5.228 

7.029 

9.652 

11.70 

13.43 

14.97 

16.36 

16. 

3.044 

5.439 

7.299 

10.01 

12.13 

13.93 

15.52 

16.97 

17. 

3.184 

5.645 

7.563 

10.37 

12.56 

U.42 

16.06 

17.55 

18. 

3.328 

5.846 

7.820' 

10.71 

12.97 

4.89 

16.58 

18.13 

19. 

3.457 

6.166 

8.069 

11.04 

13.37 

5.35 

17.09 

18.68 

f- 

3.590 

6.361 

8.313 

11.37 

13.76 

5.79 

17.59 

19.22 

. 

3.720 

6.551 

8.551 

11.63 

14.14 

6.23 

18.07 

19.75 

22%-  •  • 

3.848 

6.737 

8.783 

12.00 

14.51 

6.65 

18.54 

20.26 

23. 

3.974 

6.918 

9.010 

14.87 

7.06 

19.01 

24. 

4.098 

7.096 

9.233 

15.23 

7.47 

19.46 

25. 

4.220 

7.271 

9.451 

12.83 

15.58 

7.87 

19.95 

50. 

6.843 

10.89     ' 

13.95 

100. 

10.79 

16.07 

20.36 

-" 

220 


PRACTICAL    HYDRAULICS, 

TABLE  27. 


Flow  of  Water  per  Second  in   Open  Streams,  the  Coefficient 
of  Roughness  of  whose  beds  is  n=.O12. 


^  flfffl 

- 

*D 

II    0-0. 

II  03 

w 

»  o  '°  •*" 

$J 

lisS 

l||b 

*$ 

^   ^   O  o 

*.  2.  "o  to 

.25 
.3 

1.241 
1.423 

1.330 
1.526 

1.415 
1.623 

1.494 
1.713 

1.841 
2.110 

2.132 
2.443 

3.391 

3.884 

4.919 
5.504 

.4 

1.761 

1.898 

2.007 

2.118 

2.606 

3.017 

4.762 

6.790 

.5 

2.072 

2.220 

2.359 

2.489 

3.134 

3.544 

5.625 

7.968 

.6 

2.358 

2.526 

2.685 

2.833 

3.403 

4.029 

6.388 

9.053 

.7 

2.629 

2.815 

2.991 

3.154 

3.877 

4.485 

7.280 

10.07 

.8 

2.883 

3.088 

3.257 

3.459 

4.253 

4.919 

7.794 

11.03 

.9 

3.127 

3.348 

3.556 

3.750 

4.605 

5.326 

8.385 

11.95 

1.0 

3.360 

3.596 

3.820 

4.028 

4.946 

5.719 

9.064 

12.83 

1.25 

3.902 

4.176 

4.435 

4.676 

5.738 

6.633 

10.50 

14.87 

1.5 

4.403 

4.710 

5.001 

5.272 

6.468 

7.475 

11.81 

16.74 

2. 

5.302 

5.671 

6.018 

6.344 

7.778 

8.986 

14.22 

20.11 

2.5 

6.107 

6.531 

6.851 

7.304 

8.951 

10.33 

16.35 

23.13 

3. 

6.843 

7.317 

7.674 

8.180 

10.02 

11.84 

18.30 

25.88 

3.5 

7.525 

8.044 

8.436 

8.992 

11.01 

12.71 

20.10 

28.43 

4. 

8.165 

8.727 

9.151 

9.153 

11.94 

13.78 

21.79 

30.81 

4.5 

8.769 

9.372 

9.826 

10.47 

12.82 

14.80 

23.39 

33.07 

5. 

9.341 

9.983 

10.46 

11.15 

13.65 

15.75 

24.90 

35.20 

5.5 

9.888 

10.56 

11.20 

11.80 

14.44 

16.67 

26.34 

37.24 

6. 

10.41 

11.13 

11.79 

12.44 

15.21 

17.55 

27.73 

39.19 

6.5 

10.91 

11.66 

12.36 

13.02 

15.94 

18.39 

29.05 

41.07 

7. 

11.40 

12.17 

12.91 

13.60 

16.64 

19.20 

30.32 

42.86 

7.5 

11.87 

12.68 

13.44 

14.16 

17.32 

19.99 

31.57 

44.62 

8. 

12.32 

13.16 

13.96 

14.70 

17.98 

20.75 

32.77 

46.31 

8.5 

12.76 

13.63 

14.45 

15.23 

18.62 

21.43 

33.93 



9. 

13.19 

14.09 

14.94 

15.73 

19.24 

22.20 

35.05 

9.5 

13.61 

14.54 

15.41 

16.23 

19.85 

22.90 

10. 

14.02 

14.97 

15.87 

16.71 

20.44 

23.58 

11. 

14.81 

15.81 

16.76 

17.64 

21.58 

12. 

15.56 

16.61 

17.61 

18.54 

22.67 



13. 

16.28 

17.38 

18.42 

19.41 

14. 

16.97 

18.13 

19.21 

20.28 

15. 

17.64 

18.84 

19.97 

16. 

18.29 

19.53 

20.71 

17. 

18.93 

28.21 

18. 

19.54 

19. 

20.14 

PRACTICAL   HYDRAULICS. 


221 


TABLE  27. 

Flow  of  Water  per  Second  in  Open  Streams,  the  Coefficient 
of  Roughness  of  whose  beds  is  n=.O17. 


Hydraulic 
Mean  Depth, 
T==_a_ 
P 

,*">? 

lid 

•32% 

F=.264. 
«=.  00005. 
Velocity. 
Feet. 

<!<*,  w 

?!!  .» 

»HP 

<*%,  *i 

stiJ- 

»  g.oo 

<}co  Hj 

gfsfbr' 

^»? 

»TJ  <L   II      II 

rt>  o^     to 

S-2.2M 

'  ?Z$ 

*21? 

|§8^ 

^s? 

<J»  TJ 

•nan  II 
ggbs* 

~$f$ 

.25 

.0567 

.1704 

.2615 

.3956 

.5004 

.5844 

.6597 

.7288 

.3 

.0675 

.2000 

.3051 

.4595 

.5800 

.6769 

.7635 

.8412 

.4 

.0887 

.2568 

.3877 

.5796 

.7296 

.8499 

.9577 

.054 

.5 

.1095 

.3110 

.4657 

.6922 

.8691 

.011 

1.138 

.2f6 

.6 

.1301 

.3628 

.5272 

.7979 

.9997 

.162 

1.307 

.438 

.7 

.1503 

.4127 

.6101 

.8985 

1.123 

.305 

1.467 

.612 

.8 

.1702 

.4612 

.6778 

.9951 

1.239 

.441 

1.619 

.781 

.9 

.1892 

.5079 

.7423 

1  086 

1.355 

.572 

1.766 

1.986 

1. 

.2093 

.5535 

.8(166 

1.176 

1.465 

.698 

1.907 

2.095 

1.25 

.2569 

.6625 

.9565 

1.386 

1.722 

.989 

2.237 

2.456 

1.5 

.3034 

.7659 

1.097 

1.581 

1.957 

2.269 

2.544 

2.792 

2.0 

.3934 

.9581 

1.356 

1.939 

2.398 

2.768 

3.109 

3.402 

2.5 

.4802 

1.136 

1.592 

2.338 

2.793 

3.221 

3.605 

3.952 

3. 

.5629 

1.302 

1.812 

2.563 

3.157 

3.808 

4.069 

4.458 

3.5 

.6457 

1.460 

2.018 

2.843 

3.496 

4.025 

4.506 

4.929 

4. 

.7253 

1.609 

2.213 

3.100 

3.816 

4.390 

4.906 

5.372 

4.5 

.8030 

1.753 

2.455 

3.358 

4.119 

4.735 

5.291 

5.792 

5. 

.8590 

1.890 

2.576 

3.594 

4.407 

5.064 

5.657 

6.191 

5.5 

.9533 

2.023 

2.745 

3.824 

4.683 

5.379 

6.007 

6.574 

6. 

1.026 

2.101 

2.910 

4.044 

4.943 

5.682 

6.344 

6.941 

6.5 

1.098 

2.275 

3.C68 

4.256 

5.204 

5.973 

6.667 

7.134 

7. 

1.168 

2.394 

3.220 

4.460 

5.450 

6.253 

6.978 

7.633 

7.5 

1.238 

2.512 

3.361 

4.659 

5.690 

6.526 

7.282 

7.964 

8. 

1.307 

2.582 

3.433 

4.851 

5.921 

6.790 

7.575 

8.280 

8.5 

1.373 

2.740 

3.570 

5.038 

6.147 

7.(i47 

7.861 

8.595 

9. 

1.439 

2.845 

3.790 

5.220 

6.365 

7.296 

8.137 

8.896 

9.5 

1.505 

2.951 

3.923 

5.397 

6.579 

7.540 

8.407 

9.191 

10. 

1.569 

3.055 

4.054 

5.444 

6.788 

7.777 

8.673 

9.479 

11. 

1.696 

3.256 

4.307 

6.044 

7.192 

8.256 

9.182 

10.03 

12. 

1.820 

3.519 

4.539 

6.226 

7.572 

8.878 

9.669 

10.56 

13. 

1.939 

3.638 

4.783 

6.537 

7.950 

9.100 

10.14 

11.08 

14. 

2.059 

3.819 

5.009 

6.680 

8.309 

9.487 

10.59 

11.57 

15. 

2.175 

3.994 

5.257 

7.125 

8.654 

9.901 

11.03 

12.05 

16. 

2.289 

4.165 

5.438 

7.404 

8.990 

10.28 

11.45 

12.51 

17. 

2.401 

4.331 

5.644 

7.675 

9.317 

10.65 

11.86 

12.96 

18. 

2.511 

4.493 

5.846 

7.940 

9.635 

11.01 

12.26 

13.39 

19. 

2.619 

4.651 

6.181 

8.196 

9.943 

11.36 

12.65     . 

13.82 

20. 

2.726 

4.806 

6.231 

8.448 

10.24 

11.71 

13.03 

14.23 

21. 

2.836 

4.957 

6.417 

8.694 

10.54 

12.04 

13.40 

14.64 

22. 

2.935 

5.096 

6.598 

8.933 

10.82 

12.37 

13.77 

15.03 

23. 

3.036 

5.249 

6.777 

9.273 

11.11 

12.69 

14.12 

15.42 

24. 

3.137 

5.391 

6.952 

9.397 

11.38 

13.  CO 

14.47 

15.80 

25. 

3.237 

5.531 

7.123 

9.622 

11.65 

13.31 

14.81 

16.17 

50. 

6.402 

8.439 

10.67 

14.27 

17.22 

19.64 

21.83 

100. 

8.766 

12.64 

15.75 

9.0.90 

222 


PRACTICAL   HYDRAULICS. 


TABLE  27. 


FJow  of  Water  per  Second  in  Open  Streams,  the  Coefficient 
of  Roughness  of  whose  beds  is  n=.O17. 


Hydraulic 
Mean  Depth, 
v—a 

F=3.696 
8=,0007. 
Velocity. 
Feet. 

<!»F 

*Tj£.JI      II 

*glr 

!§8^ 
r*8g 

,ITT 

as  0  -ui 

*ll« 

•<•»  ^ 

Its 

'     «<    W.tO 

<!»^ 

*4  £.  II   II 

G>    O   '      M 

rfPs 

*j|f  ? 

!  S-88 
-  *•  f* 

•^00    *J 

II-1  i 

g.  Q.  o  to 

•,fr-  » 

.25 
.3 
.4 
.5 
.6 
.7 
.8 
.9 
1. 
1.25 
1.5 
2. 
2.5 
3. 
3.5 
4. 
1.5 
5. 
5.5 
6. 
8.5 
7. 
7.5 
B. 
3  5 

.7893 
.9128 
1.143 
1.358 
1.558 
1.748 
1.928 
2.102 
2.268 
2.658 
3.020 
3.678 
4.271 
4.817 
5.324 
5.802 
6.255 
6.684 
7.096 
7.491 
7.871 
8.212 
8.594 
8.938 
9  166 

.8463 
.9785 
1.200 
1.455 
1.708 
1.871 
2.064 
2.249 
2.427 
2.654 
3.231 
3.931 
4.567 
5.149 
5.823 
6.200 
6.683 
7.141 
7.583 
8.003 
8.408 
8.797 
9.178 
9.545 
9  902 

.9002 
1.039 
1.303 
1.546 
1.773 
1.988 
2.193 
2.445 
2.578 
3.020 
3.430 
4.174 
4.844 
5.462 
6.036 
6.526 
7.086 
7.572 
8.036 
8.483 
8.912 
9.324 
9.683 
10.12 
10  49 

.9508 
1.074 
1.375 
1.632 
1.872 
2.098 
2.314 
2.521 
2.720 
3.185 
3.618 
4.401 
5.108 
5.758 
6.362 
6.931 
7.469 
7.979 
8.469 
8.7?5 
9.390 
9.824 
10.25 
10.66 
11  05 

1.172 
1.354 
1.693 
2.009 
2.302 
2.580 
2.844 
3.098 
3.344 
3.911 
4.543 
5.347 
6.260 
7.053 
7.790 
8.484 
9.141 
9^164 
1TO6 
10.93 
11.48 
12.01 
12.53 
13.01 
13  51 

1.358 
1.568 
1.961 
2.325 
2.664 
2.985 
3.290 
3.568 
3.863 
4.521 
5.130 
6.540 
7.231 
8.145 
8.995 
9.794 
10.95 
11.26 
11.95 
32.61 
13.25 
13.86 
14.45 
15.03 
15  23 

2.160 
2.494 
3.116 

3.eaL 

4.2HF 

5  '219 
5.680 
6.124 
7.163 
8.143 
9.  869 
11.44 
12.88 
14.22 
15.48 
16.67 
17.81 
18.88 
19.93 
20.93 
21.88 
22.82 
23.73 
24  61 

3.C61 
3.534 
4.414 
5.230 
5.9S8 
6.707 
7.474 
8.042 
8.669 
10.14 
11.44 
13.96 
16.18 
18.22 
20.11 
21.89 
23.57 
25.17 
26.70 
28.17 
29.48 
30.94 
32.26 
33.54 

9 

9  597 

10  25 

10  86 

11  44 

13  98 

16  13 

25.46 

9  5 

9  914 

10  59 

11  22 

11  82 

14  44 

16  66 

) 

10  22 

10  91 

11  57 

12  18 

14  89 

17.18 

10  82 

11  55 

12  24 

12^93 

15  76 

18  18 

2 

11  39 

12  16 

12  89 

13*67 

16  58 

19.12 

11  95 

12  75 

13  51 

14  23 

•  17  38 

20  00 

1 

12  48 

13  32 

14  11 

14  86 

18  10 

>     ' 

13  99 

13  86 

14  69 

15  47 

18.89 

5'  *. 

13  49 

14  36 

15  25 

16  06 

19  61 

f 

13  97 

14  91 

15  79 

16  64 

20.31 

j  '*. 

14  44 

15  41 

16  32 

17  19 

| 

14  89 

15  90 

16  80 

17  73 

o 

15  34 

16  33 

17  34 

18  26 

1 

15  78 

16  82 

17  83 

18  78 

2 

16  20 

17.29 

18.31 

19.28 

3 

16  62 

17  73 

18  78 

19  77 

1 

1  7  03 

18  17 

19  24 

5. 

17.43 

18.59 

19.69 

PRACTICAL   HYDRAULICS. 


223 


TABLE  27. 

Flow  of  Water  per  Second  in  Open  Streams,  the  Coefficient 
of  Roughness  of  whose  beds  is  n=.O25. 


Hydraulic 
Mean  Depth, 

r=-£- 

p 

«tl? 

*§.§§ 

*?S5 

v& 

«  2.§J« 

NfSB 

<!!!*> 

ifi1 

*|ip 

<<*>  «3 
*j2j   II 

w 

-3*>  *4 

iHi 

<*-?;•  §01 

'  ^  ^^ 

^11  ? 

bcj  0,  N  .|| 
n  o  ®  -° 

•  vT.^K 

<^*a 
igg  &  fl  II 

a  o  g» 

*4pS£ 

*2T? 

Is-S* 

r4§i 

.25 

.0365 

.1022 

.1584 

.2394 

.3031 

.3528 

....3978 

.4387 

.3 

.0435 

.1206 

.1862 

.2802 

.3519 

.4142 

.4642 

.5117 

.4 

.0624 

.1564 

.2394 

.3580 

.4483 

.5238 

-.5899 

.6498 

.5 

.0712 

.1909 

.2901 

.4316 

.5393 

.6296 

-.7086 

.7800 

.6 

.0848 

.2243 

.3388 

.5017 

.6258 

.7296 

,8208 

.9027 

.7 

.0982 

.2567 

.3858 

.5691 

.7089 

.8256 

.9282 

1.020 

.8 

.1115 

.2884 

.4311 

.6341 

.7887 

.9180 

1.031 

1.134 

.9 

.1247 

.3192 

.4761 

.6966 

.8653 

1.006 

1.130 

1.243 

1. 

.1378 

.3494 

'.5183 

.7576 

.9400 

1.093 

1.226 

1.348 

1.25 

.1662 

.4223 

.6211 

.9025 

1.117 

1.297 

1.457     ' 

1.598 

1.5 

.2017 

.4921 

.7186 

1.039 

1.283 

1.488 

1.669 

1<832 

2. 

.2636 

.6239 

.9006 

1.291 

1.590 

1.841 

2.06? 

2.264 

2.5 

.3235 

.7134 

1.079 

1.514 

1.872 

2.164 

2.422 

2.655 

3. 

.3826 

.8641 

1.227 

1.739 

2.132 

2.464 

2.755 

3.020 

3.5 

.4399 

.9758 

1.377 

1.943 

2.303 

;2.746 

3.068 

3.361 

4. 

.4965 

1.083 

1.519 

2.136 

2.610 

3.010 

3.371 

3.684 

4.5 

.5518 

1.214 

1.656 

2.321 

2.832 

3.264 

3.645 

3.993 

5. 

.6062 

1.290 

1.7S6 

2.496 

3.045 

3.506 

3.921 

4.279 

5.5 

.6600 

1.382 

1.913 

2.666 

3.247 

3.738 

4.161 

4.566 

6. 

.7128 

1.476 

2.034 

2.831 

3.443 

3.962 

4.420 

4.838 

6.5 

.7650 

1.567 

2.153 

2.988 

3.634 

4.180 

4.662 

5.100 

7. 

.8165 

1.655 

2.267 

3.141 

3.815 

4.388 

4.893 

5.352 

7.5 

.8671 

1.741 

2.379 

3.291 

3.996 

4.592 

5.121 

5.599 

8. 

.9202 

1.826 

2.488 

3.433 

4.169 

4.788 

5.340 

5.840 

8.5 

.9607 

1.909 

2.594 

3.575 

4.337 

4.982 

5.552 

6.070 

9. 

.016 

1.990 

2.697 

3.712 

4.500 

5.168 

5.760 

6.295 

9.5 

.065 

2.069 

2:790 

3.846 

4.661 

5.352 

5.961 

6.518 

10. 

.113 

2.146 

2.898 

3.978 

4.818 

5.530 

6.160 

6.734 

11. 

.208 

2.353 

3.091 

4.357 

5.101 

6.056   , 

6.568 

7.131 

12. 

.299 

2.444 

3.276 

4.491 

5.413 

6.208 

6.911 

7.552 

1:5. 

.392 

2.583 

3.455 

4.714 

5.695 

6.530 

7.267 

7.939 

14. 

.482 

2.724 

3.629 

4.941 

5.967 

6.838 

7.612 

8.311 

15. 

1.569 

2.793 

3.797 

5.162 

6.228 

7.136 

7.942 

8.671 

16. 

1.658 

2.988 

3.960 

5.375 

6.484 

7.426 

8.262 

9.021 

17. 

1.742 

3.115 

4.127   . 

5.585 

6.732 

7.708 

8.575 

9.362 

18. 

1.828 

3.241 

4.278 

5.788 

6.975 

7.986 

8.879 

9.695 

19. 

1.915 

3.362 

4.419 

5.985 

7.209 

8.272 

9.186 

10.02 

20. 

1.992 

3.480 

4.514 

6.178 

7.439 

8.532 

9.488 

10.33 

21. 

2.074 

3.598 

4.719 

6.368 

7.647 

8.770 

9.749 

10.68 

22. 

2.155 

3.712 

4.862 

6.553 

7.884 

9.020 

10.02 

10.95 

23. 

2.234 

3.824 

5.000 

6.734 

8.018 

9.264 

10.30 

11.24 

24. 

2.312 

3.934 

5.150 

6.911 

8.310 

9.502 

10.56 

11.52 

25. 

2.390 

4.052 

5.270 

;7.085 

8.516 

9.738 

10.82 

11.81 

50. 

4.127 

6.325 

8.064 

10.70 

12.80 

14.61 

16.21 

17.67 

100. 

6.916 

9.673 

12.00 

15.89 

18.94... 

21.58 

224 


PRACTICAL   HYDRAULICS. 


TABLE  27. 


Plow  of  Water  per  Second  in  Open  Streams,  the  Coefficient 
of  Roughness  of  whose  beds  is  n=.O25. 


-If 

-tjdJI   ll_ 

•<<??  "9 
£o'    1 

j  GO    *^ 

lll^ 

$11 

<<*,  "3 
Kl'l 

nil 

nil 

*lsl« 

?«s 

^  '    • 

9.   * 

<<  .  * 

'   C^po 

.25 

.4759 

.5116 

.5457 

.5765 

.7072 

.8197 

1.301 

1.847 

.3 

.5551 

.5951 

.6330 

.6847 

.8238 

.9544 

1.515 

2.150 

.4 

.7045 

.7548 

.8028 

.8491 

.044 

1.209 

1.918 

2.722 

.5 

.8318 

.8986 

.9630 

1.019 

.254 

1.450 

2.299 

3.259 

.6 

.9787 

1.048 

1.114 

1.177 

.446 

1.674 

2.655 

3.765 

.7 

1.106 

1.184 

1.259 

1.330 

.633 

1.890 

2.996 

4.248 

.8 

1.228 

1.315 

1.398 

1.477 

.813 

2.097 

3.324 

4.711 

.9 

1.345 

1.440 

1.530 

1.616 

.984 

2.295 

3.637 

5.1-4 

1. 

1.459 

1.564 

1.664 

1.755 

2.151 

2.4S8 

3.941 

5.585 

1.25 

1.729 

1.855 

1.980 

2.092 

2.545 

2.943 

4.660 

6.600 

1.5 

1.981 

2.127 

2.265 

2.383 

2.914 

3.368 

5.333 

7.551 

2. 

2.446 

2.621 

2.787 

2.934 

3.590 

4.143 

6.418 

9.290 

2.5 

2.804 

3.035 

3.261 

3.434 

4.206 

4.861 

8.693 

10.90 

3. 

3.262 

3.487 

3.702 

3.899 

4.775 

5.514 

8.719 

12.31 

3.5 

3.630 

3.880 

4.113 

4.237 

5.310 

6.131 

9.694 

13.71 

4. 

3.979 

4.24S 

4.503 

4.746 

5.817 

6.712 

10.61 

15.04 

4.5 

4.310 

4.599 

4.872 

5.139 

6.297 

7.272 

11.49 

16.24 

5. 

4.625 

4.936 

5.223 

5.509 

6.754 

7.795 

12.32 

17.41 

5.5 

4.929 

5.255 

5.559 

5.866 

7.192 

8.300 

13.11 

18.53 

6. 

5.220 

5.566 

5.886 

6.211 

7.614 

8.78$ 

13.89 

19.62 

6.5 

5.503 

5.864 

6.198 

6.543 

8.025 

9.257 

14.62 

20.68 

7. 

5.773 

6.152 

6.501 

6.862 

8.416 

9.709 

15.33 

21.67 

7.5 

6.640 

6.432 

6.795 

7.178 

8.803 

10.15 

16.03 

22.65 

8. 

6.297 

6.703 

7.080 

7.485 

9.194 

10.59 

16.71 

23.60 

8.5 

6.545 

6.966 

7.356 

7.773 

9.527 

10.91 

17.16 

24.13 

9. 

6.789 

7.224 

7.626 

8.054 

9.884 

11.40 

17.96 

25.42 

9.5 

7.001 

7.461 

7.890 

8  .  332 

10  23 

11  79 

18  .  62 

10. 

7.260 

7.721 

8.148 

8.601 

10  .'53 

12!  18 

19.24 

11. 

7.948 

8.454 

8.919 

9.306 

11.21 

12.89 

20.39 



12. 

8.141 

8.655 

9.126 

9.642 

11.83 

13.64 

21.49 

13. 

8.556 

9.090 

9.585 

10.13 

K.43 

14.33 

22.62 

14. 

8.937 

9.503 

10.03 

10.59 

13!  02 

15  .  or 

15. 

9.318 

9.982 

10.46 

11.05 

13.57 

15.64 

16. 

9.723 

10.32 

10.87 

11.49 

14.11 

16.27 

17. 

10.09 

10.71 

11.28 

11.92 

14.64 

16.88 

18. 

10.44 

11.08 

11.67 

12.33 

15.16 

17.47 

19. 

10.79 

11.45 

12.05 

12.75 

15.66 

18.05 

20. 

11.10 

11.81 

12.43 

13.14 

16  .  15 

18.61 

21. 

11.46 

12.16 

12.80 

13.53 

16.61 

19.16 

22. 

11.78 

12.48 

13.15 

13.91 

17.09 

19.70 

23. 

12.10 

12.82 

13.50 

14.27 

17.55 

20.22 

24. 

12.41 

13.15 

13.84 

14.53 

17.99 

25. 

12.72 

13.48 

14.18 

15.00 

18.43 

50  ! 

19  '.01 

2o!  13 

PRACTICAL    HYDRAULICS. 


225 


TABLE  27. 


Flow  of  Water  per  Second  in  Open  Streams,  the  Coefficient 
of  Roughness  of  whose  beds  is  n=.O35. 


*H 

<1*  NJ 

^ 

~ 

~^ 

^ 

^*>*l 

~~ 

~ 

**     ^  ^ 

^y,        ft*        J'                    || 

^2  ii  if 

*T\  3-    11       il 

hel  £L  11    11 

w  ^L  11   II 

i-ri  <£•'  1!     11 

*2-l.|. 

*tl! 

?|l| 

*ffi 

So  gin 

-tii 

^ili 

*f|| 

S4IP 

j?^' 

.25 

.0251 

.0674 

.1033 

.1551 

.1947 

.2279 

.2565 

.2832 

.3 

.0300 

.0799 

.1219 

.1825 

.2288 

.2676 

.3011 

.3322 

.4 

.0397 

.1040 

.1545 

.23^3 

.2944 

.3439 

.3867 

.4215 

.5 

.0493 

.1280 

.1929 

.2860 

.3571 

.4166 

.4682 

.5160 

.6 

.0588 

.1546 

.2266 

.3347 

.4171 

.4862 

.5462 

.6015 

.7 

.0683 

.1737 

.2594 

.3817 

.4752 

.5547 

.6215 

.6841 

.8 

.0771 

.1959 

.2944 

.4274 

.  5313 

.6184 

.6941 

.7628 

.9 

.0870 

.2177 

.3225 

.4718 

.5858 

.6815 

.7648 

.8412 

1. 

.0963 

.2391 

.3555 

.5152 

.6389 

.7429 

.8334 

.9165 

1.25 

.1192 

.2911 

.4267 

.6212 

.76ri3 

.8898 

.9976 

1.096 

1.5 

.1418 

.3413 

.4972 

.7521 

.8872 

1.029 

1.153 

1.266 

2. 

.2160 

.4372 

.6303 

.9035 

1.112 

1.288 

1.442 

1.583 

2.5 

.2298 

.5283 

.  7542 

1.076 

1.322 

1  .  528 

1.711 

1>75 

3. 

.2727 

.6297 

.8734 

1.296 

1.518 

1.753 

1.961 

2.100 

3.5 

.3147 

.6991 

.9861 

1.392 

1.704 

1.966 

2.198 

2.407 

4. 

.3562 

.7800 

1.094 

1.539 

1.881 

2.169 

2.423 

2.653 

4.5 

.3971 

.8583 

1.198 

1.6SO 

2.050 

2.363 

2.638 

2.888 

5. 

.4374 

.9343 

1.296 

1.815 

2.213 

2.543 

2.846 

3.113 

5.5 

.4774 

1.008 

1.396 

1.946 

2.370 

2.728 

3.045 

3.331 

6. 

.5168 

1.081 

1.490 

2.075 

2.464 

2.902 

3.238 

3.540 

6.5 

.5558 

1.151 

1.583 

2.195 

2.669 

3.069 

3.425 

3.744 

7. 

.  5944 

.219 

1.671 

2.314 

2.811 

3.232 

3.605 

3.940 

7.5 

.6327 

.287 

1.759 

2.431 

2.971 

3.391 

3.782 

4.132 

8. 

.6706 

.353 

1.802 

2.544 

3.0S6 

3.545 

3.953 

4.319 

8.5 

.7080 

.418 

1.927 

2.655 

3.218 

3.695 

4.120 

4.501 

9. 

.7454 

.482 

2.009 

2.763 

3.347 

3.843 

4.284 

4.678 

9.5 

.7823 

.514 

2.089 

2.868 

3.473 

3.986 

4.444 

4.851 

10. 

.8189 

.605 

2.163 

2.972 

3.597 

4.127 

4.600 

5.022 

11. 

.8911 

.726 

2.315 

3.173 

3.837 

4.401 

4.904 

5.351 

12. 

.9625 

.842 

2.463 

3.368 

4.068 

4.663 

5.195 

5.668 

13. 

1.033 

.958 

2.612 

3.556 

4.204 

4.917 

5.477 

5.974 

14. 

1.102 

2.063 

2.751 

3.738 

4.508 

5.163 

5.750 

6.271 

15. 

1.171 

2.174 

2.886 

3.914 

4.717 

5.401 

6.014 

6.557 

16. 

1.238 

2.279 

3.017 

4.086 

4.921 

5.633 

6.271 

6.836 

17. 

1.305 

2.382 

3.146 

4.253 

5.120 

5.858 

6.500 

7.108 

18. 

1.372 

2.483 

3.271 

4.  489 

5.314 

6.079 

6.765 

7.373 

19. 

1.437 

2.582 

3.394 

4.576 

5.502 

6.293 

7.003 

7.631 

20. 

1.501 

2.679 

3.514 

4.732 

5.688 

6.503 

7.235 

7.883 

21. 

1.569 

2.774 

3.632 

4.885 

5.869 

6.709 

7.463 

8.131 

22. 

1.629 

2.867 

3.747 

5.033 

6.046 

6.909 

7.6S2 

8.372 

23. 

1.692 

2.951 

3.861 

5.180 

6.228 

7.105 

7.903 

8.607 

24. 

1.754 

3.049 

3.972 

5.324 

6.398 

7.299 

8.118 

8.840 

25. 

1.816 

3.139 

4.0S1 

5.465 

6.556 

7.488 

8.327 

9.  068 

50. 

3.223 

5.038 

6.385 

8.420 

10.04 

11.44 

12.70 

13.81 

100. 

5.825 

7.878 

9.991 

12.71 

1  .10 

17.15 

19.03 

20.68 

226 


PRACTICAL   HYDRAULICS. 


TABLE  27. 


Flow  of  Water  per  Second  in  Open  Streams,  the  Coefficient 
of  Roughness  of  whose  beds  is  n=.O35. 


IT    * 

**  •»  **t 

^  £.  jl  Ij 

||?l 

M 

ill? 

|f»» 

fill 

nil 

*j2.^f  | 

$.  2.  o  to  ' 

*|*|| 

Ur^i 

irsp 

NjB. 

&    p 

t.  - 

«<"  ^  bo 

.25 

.3071 

.3294 

.3503 

.3700 

.4562 

.5163 

.8401 

1.190 

.3 

.3603 

.3365 

.4108 

.4339 

.  5345 

.6191 

.9843 

1.395 

.4 

.4621 

.4954 

.5266 

.5561 

.6346 

.7927 

1.259 

1.784 

.5 

.5590 

.5992 

.6368 

.6723 

.8273 

.9577 

1.521 

2.155 

.6 

.6517 

.6983 

.7419 

.7832 

.9634 

1.115  v 

1.770 

2.507 

.7 

.  7408 

.7937 

.8241 

.8901 

1.094 

1.266 

2.010 

2.846 

.8 

.8270 

.8858 

.9411 

.9932 

1.221 

1.412 

2.240 

3.173 

.9 

.9107 

.97f>3 

1.036 

1.097 

1.343 

1.555 

2.464 

3.488 

1. 

.9920 

1.062 

1.128 

1.190 

1.482 

1.691 

2.682 

3.797 

1.25 

1.186 

1.270 

1.343 

1.422 

1.734 

2.019 

3.200 

4.529 

1.5 

1.370 

1.468 

1.556 

1.64/1 

2.615 

2.329 

3.6S9 

5.221 

2. 

1.672 

1.330 

1.943 

2.049 

2.512 

2.903 

4.596 

6.502 

2.5 

2.026 

2.167 

2.299 

2.424 

2.972 

3.433 

5.430 

7.632 

3. 

2.393 

2.482 

2.643 

2.650 

3.400 

3.927 

6.201 

8.783 

3.5 

2.600 

2.779 

2.947 

3.107 

3.804 

4.393 

6.945 

9.820 

4. 

2.865 

3.061 

3.247 

3.422 

4.188 

4.836 

7.642 

10.80 

4.5 

3.118 

3.331 

3.543 

3.723 

4.556 

5.270 

8.307 

11.74 

5. 

3.361 

3.590 

3.807 

4.011 

4.907 

5.663 

8.945 

12.65 

5.5 

3.594 

3.840 

4.070 

4.239 

5.246 

5.928 

9.559 

13.57 

6. 

3.820 

4.081 

4.325 

4.557 

5.573 

6.429 

10.15 

14.35 

6.5 

4.039 

4.314 

4.572 

4.817 

5.889 

6.793 

10.72 

15.16 

7. 

4.250 

4.539 

4.810 

5.067 

6.194 

7.145 

11.28 

15.94 

7.5 

4.459 

4.759 

5.044 

5.313 

6.493 

7.490 

11.81 

16.70 

8. 

4.657 

4.973 

5.269 

5.551 

6.783 

7.823 

12.34 

17.44 

8.5 

4.853 

5.181 

5.490 

5.783 

7.065 

8.148 

12.85 

18.16 

9. 

5.044 

5.372 

5.705 

6.008 

7.340 

8.475 

13.05 

18.87 

9.5 

5.231 

5.584 

5.916 

6.230 

7.610 

8.775 

13.84 

19.56 

10. 

5.413 

5.778 

6.121 

6.447 

7.855 

9.079 

14.32 

20.23 

11. 

5.781 

6.155 

6.521 

6.8B7 

8.385 

9.  668 

15.24 

12. 

6.108 

6.504 

6.904 

7.269 

8  .  674 

10.47 

16.13 

13. 

6.438 

6.740 

7.227 

7.661 

9.538 

10.78 

17.03 

14. 

6.757 

7.209 

7.635 

8.033 

9.811 

11.30 

17.80 

15. 

7.064 

7.537 

7.982 

8.403 

10.25 

11.82 

18.62 

16. 

7.365 

7.856 

8.320 

8.759 

10.69 

12.31 

19.40 

17. 

7.657 

8.167 

8.643 

9.104 

11.11 

12.80 

20.16 

18. 

7.942 

8.471 

8.970 

9.442 

11.52 

13.27 

19. 

8.031 

8.763 

9.231 

9.769 

11.92 

13.71 

20. 

8.489 

8.849 

9.58fl 

10.09 

12.30 

14.18 

21. 

8.755 

9  125 

9.886 

10.4* 

12.69 

14.61 

22. 

9.015 

9.614 

10.18 

10.71 

13.06 

15.05 

23. 

9.263 

9.884 

10.46 

11.01 

13.43 

15.47 

24. 

9.518 

10.15 

10.74 

11.31 

13.78 

15.88 

25. 

9.763 

10.41 

11.02 

11.60 

14.14 

16.28 

50. 

14.86 

15.84 

16.75 

17.62 

21.43 

PEACTICAL  HYDRAULICS. 


227 


TABLE   28. 


Dimensions  of  Water  Ways   Corresponding  to   Their   Given 
Hydraulic  Mean  Depths. 


M  a           Square. 

Rectangle.                     Triangle. 

Semi-circle. 

If 



V      "7 

\           7 

-TI 

II    5- 

1:1 

112 

\oy 

V_x 

II    J 

bide. 

Area. 

D'ptfc 

Width 

Area. 

Side. 

Area. 

Diam'tr 

Arei. 

i 

Feet. 

Sq.    Ft. 

Ft. 

Ft. 

Sq.  Ft. 

Feet. 

Sq.  Ft. 

Feet. 

Sq.  Ft. 

.25 

.75 

.563 

.5 

1.0 

.5 

1. 

.5 

1. 

.393 

.3 

.9 

.81 

.(i 

1.2 

.72 

1.2 

.72 

1.2 

.565 

.4 

1.2 

1.44 

.8 

1.6 

1.28 

1.6 

1.28 

1.6 

1.005 

.5 

1.5 

2.25 

1 

.0 

2.0 

2.0 

2.0 

2.00 

2. 

1.571 

.6 

1.8 

3.24 

1 

.'2 

2.4 

2.88 

.2.4 

2.88 

2.4 

2.262 

.7 

2.1 

4.41 

] 

.\ 

2.8 

3.92 

2.'8 

3.92 

2.8 

3.079 

.8 

2.4 

5.76 

] 

.(i 

3.2 

5.12 

3.2 

5.12 

3.2 

4.021 

.9 

2.7 

7.29 

] 

.S 

3.6 

'6.48 

3.6 

6.48 

3.6 

5.089 

1. 

3. 

9.00 

2.0 

4.0 

8.00 

4. 

8.00 

4. 

6.283 

1.25 

3.75 

14.06 

2.5 

5.0 

12.5 

5. 

12.5 

5. 

9.818 

1.5 

4.5 

20.25 

3.0 

6.0 

18.0 

6. 

18.0 

6. 

14.137 

2. 

6. 

36. 

4.0 

8.0 

32.0 

8. 

32.0 

8. 

25.133 

2.5 

7.5 

56.25 

5:0 

10.0 

50.0 

10. 

50.0 

10. 

39.27 

3. 

9. 

81. 

6.0 

12.0 

72.0 

12. 

72.0 

12. 

56.549 

3.5 

10.5 

110.25 

7.0 

14.0 

98.0 

14. 

98. 

14. 

76.696 

4. 

12. 

144.00 

* 

16.0 

128. 

10. 

128. 

16. 

100.53 

4.5 

13.5 

182.25 

9. 

18.0 

162. 

IS. 

162. 

18. 

127.23 

5. 

15. 

225.00 

10. 

20.0 

200. 

20. 

200. 

20. 

157.08 

5.5 

16.5 

272.25 

11. 

22.0 

242. 

22. 

242. 

22. 

190.07 

6. 

18. 

324.00 

12. 

24.0 

288. 

24. 

288. 

24^ 

226.2 

6.5 

19.5 

380.25 

13. 

26.0 

338. 

26. 

338. 

26. 

265.5 

7. 

21. 

441.00 

14. 

28.0 

392. 

28. 

392. 

28. 

307.0 

7.5 

22.5 

506.25 

15. 

30.0 

450. 

30. 

450. 

30. 

353.4 

8. 

24. 

576.00 

16. 

32.0 

512. 

32. 

512. 

32. 

402.1 

9. 

27. 

729.00 

18. 

36.0 

648. 

36. 

648. 

36. 

508.9 

10. 

30. 

900.00 

20. 

40.0 

800. 

40. 

800. 

40. 

628.3 

228 


PEACTICAL  HYDKAULICS. 


1 


00 


l\ 


II 


Hydraulic 
Mean  Depth, 


r-  r-  IH  r-  r-   <M  co  Tj  i  co  J>  t  a  e>  o  -H  < 


i  S  O5  00  kO  t-  CO  CO  00  00  Tt<  S-l  'O  •*  L^  •*  1^  -*  i-(  OO 
1H'~(  rtrHrH5-]^M^COCO-*lO<» 


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PRACTICAL   HYDKAULICS. 


229 


TABLE.  29. 


Relations  of  Depth,  Base  and  Slope  of  a  Bank  of  a   Trap- 
ezoidal Canal. 


fatio  of 
Depth 

1 

m 

1  x  m2)* 

Ratio  of 
Deoth 

1 

m 

(1+m2)* 

to  Base. 

Depth. 

Base. 

S'ope. 

to  Base. 

Depth. 

Base. 

Slope. 

5:1 

1 

.2000 

.0198 

19:4 

1 

.2105 

1.0219 

24:5 

1 

.2083 

.0215 

9:2 

1 

.2222 

1.0244 

23:5 

1 

.2174 

.0234 

17:4 

1 

.2353 

1.0272 

22:5 

1 

.2273 

.0255 

15:4 

1 

.2667 

1.0349 

21:5 

1 

.2381 

.0280 

7:2 

1 

.2856 

1.0400 

4:1 

1 

.2500 

.0306 

13:4 

1 

.3077 

1.0463 

19:5 

1 

.2632 

.0364 

11:4 

1 

.3636 

1.0641 

18:5 

1 

.2778 

.0379 

5:2 

1 

.4000 

1.0770 

17:5 

1 

.2941 

.0423  , 

9:4 

1 

.4444 

1.0940 

10:5 

1 

.3125 

.0477 

7:4 

1 

.5714 

1.1517 

3:1 

1 

.3333 

.0541 

3:2 

1 

.6667 

1  .  2046 

14:5 

1 

.3571 

.0619 

5:4 

1 

.8000 

1.2806 

13:5 

1 

.3846 

.0714 

3:4 

1 

1.3333 

1.6667 

12:5 

1 

.4167 

.0833 

1:2 

1 

2.0000 

2.2361 

11:5 

1 

.4545 

.0985 

1:4 

1 

4.0000 

4.1231 

2:1 

1 

.5000 

.1180 

14:3 

1 

.2143- 

1-0227 

9:5 

1 

.5556 

.1439 

13:3 

1 

.2308 

1.0263 

8:5 

1 

.6250 

.1793 

11:3 

1 

.2727 

1.0365 

7:5 

1 

.7143 

.2575 

10:3 

1 

.3000 

1.0440 

6:5 

1 

.8333 

.3017 

8:3 

1 

.3750 

1.0680 

1:1 

1 

l.oro 

.4142 

7:3 

1 

.4286 

1.0880 

4:5 

1 

1.250 

.6008 

5:3 

1 

.6000 

1.1662 

3:5 

1 

1.667 

.9436 

4:3 

1 

.7500 

1.2500 

2:5 

1 

2.500 

2.6926 

2:3 

1 

1.5000 

1.8028 

1:5 

1 

5.000 

5.0990 

1:3 

1 

3.0000 

3.1622 

5:24 

1 

4.8000 

4.9031 

4:19 

1 

4.7500 

4.8541 

5:23 

1 

4.6000 

4.7074 

4:18 

1 

4.5000 

4.6098 

5:22 

1 

4.4000 

4.5122 

4:17 

1 

4.2500 

4.3660 

5:21 

1 

4.2000 

4.3174 

4:15 

1 

3.7500 

3.8810 

5:19 

1 

3.8000 

3.9295 

4:14 

1 

3.5000 

3.6401 

5:18 

1 

3.6000 

3.7363 

4:13 

1 

3.2500 

3.4004 

5:17 

1 

3.4000 

3.5440 

4:11 

1 

2.7500 

2.9262 

5:16 

1 

3.2000 

3.3526 

4:9 

1 

2.2500 

2.4622 

5:14 

1 

2.8000 

2.9732 

4:7 

1 

1.7500 

2.0156 

5:13 

1 

2.6000 

2.7851 

4:6 

1 

1.5000 

1.8028 

5:12 

1 

2.4000 

2.6000 

3:14 

1 

4.6667 

4.7726 

5:11 

1 

2.2000 

2.4161 

3:13 

1 

4.3333 

4.4472 

5:9 

1 

1.8000 

2.0591 

3:11 

1 

3.6666 

3.8006 

5:8 

1 

1.6000 

3:10* 

1 

3.3333 

3.4801 

5:7 

1 

1.4000 

l!7205 

3:8 

1 

2.6667 

2.8480 

5:6 

1 

1.2000 

1.5621 

3:7 

1 

2.3333 

2.5386 

230  PRACTICAL   HYDRAULICS. 


PRACTICAL  APPLICATION  OF  TABLES  27,  28  AND  29, 

TO    DETERMINE   THE   VELOCITY   AND    DISCHARGE   OF   AN 
OPEN  STREAM  OF  WATER. 

Rule  56. — From  the  given  dimensions  of  the  stream 
find  by  Table  28  or  Table  29,  according  to  the  condi- 
tions, the  hydraulic  mean  depth.  Turn  then  to  Table 
27  for  the  given  coefficient  n,  for  roughness  of  bed. 
In  this  table,  opposite  the  ''hydraulic  mean  depth"  as 
determined,  in  the  column  headed  by  the  given  fall 
per  mile,  will  be  found  the  velocity  sought.  Multiply 
the  velocity  so  found  by  the  area  of  the  cross  section 
of  the  stream  for  the  discharge. 

Ex.  102. — The  side  of  a  square  flume  of  unplaned 
plank  is  3  feet,  the  fall  per  mile-4.752  feet  (s=.0009), 
and  the  coefficient  of  roughness  of  its  bed,  ?i=.0l2. 
(See  Table  26.)  What,  per  second,  is  the  velocity, 
and  what  the  discharge? 

Col.— In  Table  28,  find  in  "side"  column  for  a 
square  flume  or  canal.  3  feet;  opposite  which,  in  "area" 
column,  is  found  9  square  feet,  and  opposite  which,  in 
hydraulic  mean  depth  column,  is  found  1.  Turning 
now  to  Table  27,  computed  for  coefficient  of  rough- 
ness of  bed,  7i^=.012,  find  in  hydraulic  mean  depth 
column  1;  opposite  which,  in  velocity  column,  for  the 
given  fall,  F=4.752,  (s=.0009)  will  be  found  the  ve- 
locity sought,  viz. :  . 

v= 3. 82  feet. 


PRACTICAL  HYDRAULICS.  231 

Then  g  .=3. 82x9= 34. 38  cubic  feet.— Ana. 

Ex.  103. — The  depth  of  a  rectangular  canal  "being 
5  feet,  the  width  10  feet,  the  fall  per  mile  2.64  feet, 
F=2.64,  s=.0005,  and  the  coefficient  of  roughness 
of  the  bed  n-=.0l7  (see  Table  26),  what  will  be 
the  velocity  of  the  water,  and  what  the  discharge  in 
cubic  feet? 

Cal.— Find  in  Table  28,  under  ' 'rectangle"  in 
"depth"  and  "width"  columns,  the  given  depth  and 
width  5  and  10  feet;  opposite  which,  in  "area"  col- 
umn, will  be  found  50  square  feet;  and  in  the  "hy- 
draulic mean  depth  column,"  will  be  found  2.5. 

Turning  now  to  Table  27,  for  n=.0~Ll,  find  in  "hy- 
draulic mean  depth  column"  2.5;  opposite  which,  in 
velocity  column  for  the  given  fall,  F=2.64,  will  be 
found  the  velocity  sought,  viz. : 

v---=  3.605  feet. 

Then  g=3. 605X50=180.25  cubic  feet.— Ana. 

Ex.  104.— The  side  (wetted)  of  a  V  (vee)  flume  of 
unplaned  plank,  being  2.4  feet,  the  fall  per  mile  26.4 
feet,  and  the  coefficient  for  roughness  of  the  bed,  ti= 
.012  (see  Table  26),  what  is  the  velocity  of  the 
water  per  second,  and  what  the  discharge  in  cubic 
feet? 

Col. — Find  in  Table  28,  under  triangle  in  side  col- 
umn, the  given  side  2.4  feet;  opposite  which,  in  area 
column,  is  found  2.88  square  feet,  and  in  '  'hydraulic 
mean  depth  column,"  .6. 

Turning  now  to  Table  27,  for  7i^=.0l2,  find  in  "hy- 


232  PEACTICAL   HYDRAULICS. 

draulic  mean  depth"  column,  .6;  opposite  which,  in  ve- 
locity .  column  for  the  given  fall,  F=26.4,  will  be 
found  the  velocity  sought,  viz.: 

v=6. 338  feet. 

Then  £=6.338X  2.88=18.25  cubic  feet.— Ans. 

Ex.  105. — The  depth  of  a  regular  semi-hexagonal 
canal  in  earth,  being  3  feet,  the  fall  per  mile  F=5.28 
feet  (s=.00l),  and  the  coefficient  for  roughness  of  bed 
n—.Q2o  (see  Table  26),  what  is  the  velocity  of 
the  water  and  what  the  discharge? 

Col. — Find  in  Table  28,  under  semi-hexagon  in 
depth  column,  3  feet,  opposite  which,  in  area  column, 
is  found  15.588  square  feet,  and  in  hydraulic  mean 
depth  column  1.5. 

Turning  now  to  Table  27,  for  n=.025,  find  in  hy- 
draulic mean  depth  column  1.5,  opposite  which,  in 
velocity  column  for  the  given  fall,  F=5.28  feet,  will 
be  found  the  velocity  sought,  viz. : 

w=2. 383  feet. 

Then  0=15. 588X2. 383=37.15  cubic  feeb.— Ans. 

Ex.  106. — In  a  trapezoidal  canal  (bottom-slope  of 
bank),  the  angle  of  slope  of  bank  is  45°,  the  depth 
4.394  feet,  the  fall  per  mile  F=7.92  feet,  and  the  co- 
efficient for  roughness  of  bed  n=  .01 7  (see  Table  26); 
what  is  the  velocity  and  what  the  discharge  of  water 
per  second? 

Gal. — Find  in  Table  28,  under  trapezoid,  with  bank 
slope  of  45°,  in  depth  column,  4.394  feet,  the  given 


PEACTICAL  HYDRAULICS.  23b 

depth  ;  opposite  which,  in  area  column,  is  found 
46.606  square  feet,  and  in  hydraulic  mean  depth  col- 
umn, 2.5. 

Turning  to  Table  27,  find  in  hydraulic  mean  depth 
column  2.5,  opposite  which,  in  velocity  column  for 
the  given  fall,  F=7.92  feet,  is  found  the  velocity 

sought,  viz. : 

v=6.26  feet. 

Then   5=46.606x6.26=291.75  cubic  feet.— Ans. 

Ex.  107. — In  a  trapezoidal  canal,  in  which  the  bot- 
tom is  equal  a  side,  and  the  ratio  of  the  depth  to  the 
base  of  bank  is  as  2:1,  the  depth  of  water  is  2.591 
feet,  the  fall  per  mile  F=10.56  feet  (s=.002)  and  the 
coefficient  for  roughness  of  bed,  7i=.025  (see  Table  26), 
what,  per  second,  is  the  velocity  and  what  the  dis- 
charge? 

Col. — Find  in  Table  28,  under  trapezoid  2*.  1,  in 
depth  column,  the  given  depth  2.591  feet;  opposite 
which  in  area  column  is  found  10.86  square  feet,  and 
in  hydraulic  mean  depth  column  1.25. 

Turning  now  to  Table  27  for  n=  .025  find  in  hy- 
draulic mean  depth  column  1.25;  opposite  which  in 
velocity  column  for  the  given  fall  F=10.56  is  found 
the  velocity  sought,  viz. : 

v= 2.943  feet. 

Then  5=10.86X2.943=31.96  ciibic  feet.—  Ans. 

Ex.  108. — In  a  trapezoid  canal,  in  which  the  bot- 
tom is  equal  to  a  side,  and  the  ratio  of  depth  to  the 
base  of  the  bank  is  as  1:2,  the  depth  of  water"  is 


234  PRACTICAL  HYDRAULICS. 

5.543  feet,  the  fall  per  mile  1.056  feet,  and  the  coeffi- 
cient for  the  roughness  of  the  bed,  7i=.035  (see 
Table  26),  what  is  the  velocity,  and  what  the  dis- 
charge per  second  ?  - 

Cat. — Find  in  Table  28,  under  trapezoid  1-2,  in 
depth  column  the  given  depth  5.543  feet;  opposite 
which  in  area  column  is  found  130.04  square  feet, 
and  hydraulic  mean  depth  column  3.5.  Turning  now 
to  Table  27,  find  in  hydraulic  mean  depth  column  3.5; 
opposite  which  in  velocity  column  for  the  given  fall, 
F=1.056,  will  be  found  the  velocity  sought,  viz. : 

v= 1.392  feet. 

Then  9-—l30.04Xl.392-=18l.02  cubic  feet.— Ans. 

Ex.  109. — The  diameter  of  a  semi-circular  canal 
being  8  feet,  the  fall  per  mile  26.4  feet,  and  the  coeffi- 
cient for  roughness  of  bed  7i=.0l2,  (see  Table  26), 
what  is  the  velocity  per  second  and  what  the  dis- 
charge ? 

'Gal. — Find  in  Table  28,  under  semi-circle,  in  diam- 
eter column  8  feet,  the  given  diameter;  opposite  which 
in  area  column  is  found  25.133  square  feet,  and  in 
hydraulic  mean  depth  column,  2. 

Turning  now  to  Table  27,  findln  hydraulic  mean 
depth  column  2;  opposite  which  in  velocity  column 
for  the  fijiven  fall,  F=26.4  feet,  is  found  the  velocity 
sought,  viz. : 

<y--=14.22  feet. 

Then  £=25.133X14.22— 357.39  cubic  feet.— Ans. 
'Remark. — It  will  be  observed    that  Table  27,  is 


PBACTIGJLL  HYDBAULICSi  235 

equally  well  adapted  to  finding  the  flow  of  water  in 
circular  pipes  as  in  semi-circular  canals.  For  the  hy- 
draulic mean  depth  is  the  same  in  each.  Thus  in  Ex- 
ample 108,  were  it  required  to  determine  the  velocity 
of  water  in  a  circular  pipe  running  full,  the  only 
change  required  in  the  calculation  would  be  to  double 
the  area,  whence  would  occur  a  corresponding  change 
in  the  result,  as  follows: 

357.39X2=714.78  cubic  feet. 

In  case  of  foul  pipes  it  will  be  better  to  employ 
Table  27,  rather  than  Table  17,  computed  for  clean 
iron  pipes,  the  coefficient  of  roughness  for  whose  walls 
as  shown  is  71^.011. 

Ex.  110. — The  observed  data  of  a  canal  are  as  fol- 
lows: 

Width  of  bottom,  15  feet. 

Depth  of  water,  4.5  feet. 

Ratio  of  depth  to  base,  2:5. 

Fall  per  mile,  3.168  feet. 

Coefficient  of  roughness  of  bed  .017. 

What  is  the  velocity  of  flow  per  second,  and  what 
the  discharge  ? 

Gal. — Find  in  Table  29  the  given  ratio  of  depth  to 
base,  viz.:  2:5;  opposite  which  in  columns  of  depth, 
base  and  slope,  computed  for  a  depth  of  unity  are 
found  1,  2.5  and  2.6926.  Multiplying  each  by  the 
given  depth,  there  results  4.5,  11.25  and  12.1167,  the 
depth,  base  and  slope  of  bank.  The  wet  perimeter  is 


236  PEACTICAL  HYDRAULICS. 

equal  to  the  sum  of  the  bottom  and  twice  the  slope  of 
bank: 

p=  15  +  12.1167X2=39.2334  feet. 

The  mean  width  of  the  stream  is  equal  to  the  sum 
of  the  bottom  and  of  the  base  of  the  bank: 

^=15  +  11.25=26.25. 

The  area  of  cross  -section  of  stream  is  equal  to  the 
product  of  the  mean  width  and  depth  : 

a=26.25X  4.5  —  118.125  square  feet. 

The  hydraulic  mean  depth  is  equal  to  the  quotient 
arising  from  dividing  the  area  of  cross-section  by  the 
wet  perimeter: 

r=  JL=H8.125-*-  39.2334  -=3.01. 


Turning  now  to  Table  27  for  the  given  coefficient 
for  roughness  of  bed,  7i=.0l7,  find  in  hydraulic  mean 
depth  column  3,  the  nearest  approximate  to  that  de- 
termined, viz.:  3.01;  opposite  which  in  velocity  column 
for  the  given  fall  per  mile,  F=3.168,  is  found  the 
velocity  sought,  viz.: 

v=-4.458  feet. 

Then  g=-118.125X  4.458  =  526.6  cubic  feet.—  Am. 
Ex.  111.  —  The  data  for  the  Sacramento  river  being 
as  follows,  viz.: 

Fall  per  mile  .528  feet. 

Depth,  25  feet. 

Width  of  bottom,  453.33. 

Ratio  of  depth  to  base  of  bank,  5'-  12. 


PEACTICAL   HYDRAULICS.  237 

Coefficient  of  roughness  of  bed,  .025. 

What  will  be  the  velocity  per  second,  and  what  the 
discharge  ? 

Gal. — Find  in  Table  29  the  given  ratio,  5 : 12,  oppo- 
site which  are  found  1,  2.4  and  2.6,  multiplying  each 
by  the  given  depth,  25  feet,  there  results  25,  60  and 
65,  the  deptli,  base  and  slope. 

Then  wet  perimeter =453.33  +  65X2= 583.33   feet. 

Mean  width=453.33  +  60=513.33. 

Sectional  area-513.33  +  25^12,833.25  square  feet, 

Hydraulic  mean  depth=12,833. 25-^  583.33=22. 

By  Table  27,  the  velocity  for  the  hydraulic  mean 
depth,  22  in  velocity  column  for  the  given  fall  of  .528 
feet  is: 

?;=4.862  feet. 

Whence,  2=12,833.25X4.862=62395.26  cubic 
feet. — Ans. 

INTERPOLATION. 


For  the  purposes  of  interpolation  where  the  ex- 
tremes are  not  far  apart,  in  Table  27,  the  velocities, 
without  any  considerable  error  in  practice  may  be 
assumed  proportionate,  either  to  the  sines  of  slope  on 
one  hand,  or  to  the  hydraulic  mean  depths  on  the 
other. 

Ex.  112. — The  coefficient  of  roughness  of  bed  being 
%  — .012,  the  hydraulic  mean  depth  .5,  the  extreme 
sines  of  slope  s=.0015,  and  8=  .005,  what,  in  a  regu 


238  PRACTICAL    HYDRAULICS. 

lar  arithmetical  series  will    be   the  six   interpolated 
velocities  ? 

Cal.—By  Table  27,  for  7^.012,  and  for  the  given 
hydraulic  mean  depths  the  velocities  due  the  given 
slopes  are  3.134  and  5.625  feet  per  second.     Then 
(5.625— 3.134)-^7=- .356  common  difference. 

Whence,  the  interpolated  velocities  will  be  3.490, 
3.846,  4.202,  4.558,  4.914  and  5.270  feet.— Ans. 

Ex.  113. — The  coefficient  of  roughness  of  bed  being 
7i=.012,  the  given  sine  of  slope  s=.0007,  the  ex- 
tremes of  hydraulic  mean  depths  1  and  2,  what,  in  a 
regular  arithmetical  series,  will  be  the  three  inter- 
polated velocities? 

Cal.—By  Table  27,  for  ^=.012,  the  velocities  due 
the  given  hydraulic  mean  depths  1  and  2,  are  3.360 
and  5.302  feet.  Then  (5.302— 3.360)-^4=.4855  com- 
mon difference;  whence  the  interpolated  velocities  will 
be  3.845,  4.330,  and  4.816  feet.— Ans. 


MEAN  VELOCITY. 


To  find  the  mean  velocity  of  an  open  stream  of 
water,  various  devices,  as  tight  tin  tubes,  loaded  each 
at  one  end,  so  as  to  float  vertically  in  still  water,  and  as 
nearly  so  in  streams  as  the  current  will  permit — the 
Pitot  tube,  patent  logs,  etc.,  are  employed.  Wooden 
floats,  of  nearly  the  specific  gravity  of  water,  are, 
however,  mostly  used  in  common  practice.  By  these 


PRACTICAL  HYDRAULICS.  239 

the  surface  velocity  is  taken;  thence  the  mean  ve- 
locity is  determined  by  calculation  based  on  experi- 
mental data. 

Having  cliosen  a  straight  section  of  a  stream  free 
as  possible,  to  be  found  from  eddies,  roughness  and 
foulness  of  bottom,  and  having  divided  the  stream 
into  sections  parallel  with  its  course,  cast  into  it 
wooden  floats,  note  the  time  taken  by  a  float  in 
each  section  to  pass  through  a  given  distance,  as  100 
feet,  and  thence  determine  the  surface  sectional  ve- 
locity per  second.  Divide  the  sum  of  the  several  ve- 
locities of  the  floats  by  their  numbers;  the  result  will 
be  the  mean  surface  velocity  of  the  stream  per  second ; 
whence,  the  mean  velocity  for  the  mean  depth  of  the 
entire  cross  section  can  be  calculated  as  above  stated. 


CENTRAL  SURFACE  AND  CORRESPONDING  MEAN 
VELOCITY. 


Having  made  se\$eral  trials  with  floats,  as  above  de- 
scribed, let  the  greatest  velocity,  well  established  by 
any  one  of  them,  be  taken  as  the  central  surface  ve- 
locity. For  the  central  surface  velocities,  from  five- 
tenths  ( .5)  of  a  foot  to  six  (6)  feet,  the  corresponding 
mean  velocities  have,  by  the  aid  of  experimental  data, 
and  the  following  empirical  formula  of  Prony,  viz. : 

•   '  <*> 


240 


PRACTICAL   HYDEAULICS. 


been  computed,  and  the  results  arranged  in  Table  30. 
In  Eq.  (286),  v  denotes  the  mean  velocity,  and  V 
the  central  surface  velocity  of  a  stream  of  water. 


TABLE  30. 


Central  Surface  and  Corresponding  Mean  Velocities  of 
Streams. 


Central  Surface 
Velocity. 
Feet. 

Mean 
Velocity. 
Feet. 

Central  Surface 
Velocity. 
Feet. 

Mean 
Velocity. 
Fett. 

.5 

.382 

3.5 

2.852 

1.0 

.774 

4.0 

3.284 

1.5 

1.174 

4.5 

3.721 

2.0 

1.584 

5.0 

4.165 

2.5 

2.000 

5.5 

4.609 

3.0 

2.424 

6.0 

5.058 

Ex.  114. — What  is  the  quantity  of  flow  in  a  stream 
in  which  the  cross  section  is  50  square  feet  and  the 
central  surface  velocity  3  feet  per  second? 

Gal. — In  Table  30,  opposite  the  given  central  sur- 
face velocity  3  feet,  find  in  mean  velocity  column 
2.424  feet.  Then 

2.424X50=121.2  cubic  feet  per  second. — Ans. 

Rough  Approximate. — A  rough  approximate  is 
readily  found  by  taking  one-half  the  product  of  the 
surface  width,  central  depth,  and  central  velocity  of 
a  stream. 

This  rough  approximate  rule  is  based   on  the  as- 


PRACTICAL  HYDRAULICS.  241 

sumption  that  the  cross  section  of  stream  is  parabolic, 
and  the  mean  velocity  equal  to  three-fourths  (.75)  of 
the  central  surface  velocity. 

Ex.  115. — The  surface  width  of  a  stream  is  25  feet, 
the  central  depth  4  feet,  and  the  central  surface  ve- 
locity 1.5  feet,  what  is  the  flow  per  second? 

Gal— 25X4X1.5-^-2=75  cubic  feet.—  Ans. 

QUANTITY  OF  WATER  REQUIRED  FOR  VARIOUS  MINING 
PURPOSES. 

Hydraulic  Mining. — Hydraulic  mining,  properly, 
comprises  all  classes  of  mining  in  which  the  metallic 
substance  sought  is  separated  from  its  earthy  mass  or. 
matrix  by  means  of  water.  The  term,  however,  as 
employed  in  California,  is,  for  the  most  part,  restricted 
to  that  class  of  mining  in  which  a  stream  of  water  is 
projected  under  great  pressure  from  a  nozzle  against 
a  deep  gravel  deposit  or  earthy  formation  for  the  pur- 
poses of  disintegrating  the  mass,  thence  freeing  the 
gold  and  carrying  off  the  debris.  The  relation  of  the 
quantity  of  water  employed  to  that  of  material  re- 
moved varies  in  different  mines  and  in  different  parts 
of  the  same  mine. 

Duty  of  an  Inch  of  Water. — This  phrase  involves  in 
its  meaning  the  work  of  disintegration;  but  as  the 
projecting  head  is  variable  from  50  to  350  feet  and 
upward,  the  phrase  seems  to  refer  chiefly  to  that  por- 
tion of  the  work  performed  by  the  water  in  carrying 
off  the  debris  in  a  sluice,  the  grade  of  which  is  usually 


242  PBACTICAL   HYDRAULICS. 

6  inches  per  12  feet.  Experience  shows  that  the  duty 
of  a  24-hour  miner's  inch,  under  a  7-inch  head,  equiva- 
lent, as  shown  by  Table  8,  to  2,230  cubic- feet  flow  in 
twenty-four  hours,  is  in  the  lower  portions  of  certain 
mines  as  follows: 

Cubic  Yards. 

North  Bloomfield  Mine 3.5 

Milton  Mine 2.4 

Excelsior  Mine 2.0 

Gold  Run  Mine 3.5 

In  the  upper  portions  of  the  same  mines  the  duty 
of  the  miner's  inch  was  much  greater,  say  5  to  10 
cubic  yards. 

Thus,  at  the  Gold  Run  mine,  for  six  years  to  No- 
vember 1,  1881,  4,389,791  cubic  yards  were  worked 
with  1,124,367  miner's  inches  of  water;  whence  the 
duty  of  per  inch  was  3.9  cubic  yards. 

In  the  State  Engineer's  report  to  the  legislature  of 
the  State  of  California,  1880,  the  estimated  inch  duty 
is  as  follows,  viz.: 

Cubic  Yards. 

Yuba  River  Mines 3.5 

Bear  River  Mines 3.0 

American  River  Mines 4.5 

One  instance  is  brought  to  the  attention  of  the 
writer  showing  the  inch  duty  to  have  been  19  cubic 
yards. 

Drift  Mining. — Drift  mining  consists  in  excavating 
the  lower  material  of  a  gravel  mine  by  hand,  raising 
it  through  a  shaft  to  the  surface,  or  carrying  it  by 
wheelbarrows  and  cars  through  a  tunnel  to  a  dump, 
whence  it  is  shoveled  or  piped  into  a  sluice  to  be  freed 
of  its  gold,  and  thereupon  carried  off  as  debris.  A 


PRACTICAL  HYDRAULICS.  243 

the  larger  cobble,  bowlders  and  barren  blocks  of  rock 
are  usually  left  in  a  drift  mine,  and  the  material 
broken  smaller  than  in  hydraulic  mining,  a  corres- 
pondingly less  quantity  of  water  is  required  for  work- 
ing a  cubic  yard. 

The  duty  of  a  24-hour  inch  (2,230  cubic  feet)  in 
this  class  of  mining  varies  according  to  the  character 
of  the  gravel,  whether  hard  and  cemented,  clayey  or 
sandy,  from  3  to  20  cubic  yards. 

Quartz  Mining.— The  contents  of  one  ton  of  quartz, 
in  its  normal  condition  in  the  lode,  is  estimated  at 
13  cubic  feet,  and  at  20  cubic  feet  when  the  quartz  is 
broken,  as  it  usually  comes  from  the  mine.  Adopting 
the  lode  measurement  it  is  seen  that  a  cubic  yard  of 
quartz  is  27-*- 13=2.08  tons  nearly. 

Experience  shows  that  the  duty  of  a  miner's  inch  is 
as  follows: 

Duty  of  a  miner's  inch  (under  4-inch  pressure)  in 
the  reduction  and  amalgamation  of  silver  ores  in  a 
"stamp  silver  mill,"  Nevada,  3.25  cubic  yards  or  6.76 
tons;  in  the  reduction  and  amalgamation  by  riffles,  or 
copper  plate,  in  "stamp  gold  mill,"  California,  5.T8 
cubic  yards  or  12  tons. 

Duty  of  miner's  inch  (under  7-inch  pressure)  in  the 
former  case  (silver)  4.3  cubic  yards,  or  8.93  tons;  in 
the  latter  case  (gold)  6.65  cubic  yards,  or  15.88  tons. 

The  volume  of  water  to  that  of  ore  is,  in  working 
silver  ores,  Nevada,  19.5  to  1;  in  working  gold  ores, 
California,  11.1  to  1;  in  working  copper  ores,  Lake 
Superior,  20  to  1. 


244  PRACTICAL  HYDRAULICS. 

QUANTITY  OF  WATER  REQUIRED  FOR  PURPOSES  OF 
IRRIGATION. 

As  the  area  of  land  is  usually  expressed  in  denomi- 
nation of  acres,  a  convenient  unit  of  measure  for  irri- 
gating purposes  is  that  quantity  of  water  which  will 
cover  one  acre  one  inch  deep.  This  quantity  is  3,630 
cubic  feet. 

The  total  depth  of  irrigation,  as  practiced  in  Cali- 
fornia, varies  for  different  soils  and  products  from  two 
to  five  feet. 

Ex.  116. — It  is  proposed  to  irrigate  1000  acres  of 
land,  50  inches  in  depth,  in  100  days,  by  means  of  a 
canal  whose  fall  per  mile  is  to  be  1.056  feet  (s=.0002), 
coefficient  of  roughness  of  bed  7i=.0l7,  bottom  width 
equal  to  slant  width  of  side,  and  ratio  of  depth  to  base 
of  bank  as  1  •  2,  what  will  be  the  dimensions  of  the 
canal? 

CaL— 3630X50-^-100=:1815  cubic  feet;  1815 X 
1000— 1815000  cubic  feet  per  day;  1815000^86400- 
21.007  cubic  feet  per  second. 

Assume,  by  way  of  trial,  the  hydraulic  mean  depth 
to  be  1.25.  Then,  in  Table  27,  for  ™=.017,  the  ve- 
locity for  hydraulic  mean  depth  1.25  is  1.386  feet  per 
second;  whence,  21.007-^- 1.386=15.16  square  feet, 
area  of  cross  section  of  canal. 

By  Table  28,  for  hydraulic  mean  depth  1.25,  the 
sectional  area  is  16.59  square  feet,  under  trapezoid  1 '  2. 
This  approximate  is  sufficiently  near  to  meet  the  re- 
quirements of  practice.  The  dimensions  of  the  canal 


PRACTICAL  HYDBAULICS.  245 

then,  as  per  Table  28,  are:  side =bottom— 4.426  feet; 
depth=1.980  feet. — Ans. 

If  greater  accuracy  be  required,  proceed  as  in  the 
solution  of  Ex.  99. 

Ex.  117. — How  many  acres  can  be  irrigated  40 
inches  in  depth  in  75  days,  by  means  of  a  semi- 
hexagonal  canal  five  feet  deep,  the  fall  per  mile  being 
1.584  feet  (s=.0003),  and  the  coefficient  for  roughness 
of  bed  being  n  =  .025? 

Cal. — By  Table  28,  it  is  seen  that  the  hydraulic 
mean  depth  and  the  area  of  a  semi-hexagon  five  feet 
deep,  are  respectively:  2.5  feet  and  43.301  square  feet. 

By  Table  27,  for  w=.025,  fall  per  mile  1.584  feet, 
the  velocity  corresponding  to  hydraulic  mean  depth 
2.5  is  1.872  feet  per  second. 

Then  43.301X1-872=81. 059472  cubic  feet  per  sec- 
ond; 81.059472X86400X75=525265378.5  cubic  feet; 
3,630X40=145200  cubic  feet  per  acre;  525265378.5 
-*- 145200 -3617.5  acres.—  Ans. 

MEASUREMENT  OF  THE  POWER  OF  WATER  AS  A  MOTOR. 

The  unit  in  the  measurement  of  power  is  a  foot- 
pound— that  is,  the  amount  of  energy  necessary  to 
raise  one  pound  weight  vertically  through  a  distance 
of  one  foot.  On  the  other  hand,  one  pound  falling  by 
the  force  of  gravity  through  a  distance  of  one  foot, 
generates  a  foot-pound. 

The  amount  of  energy  required  to  raise  one  pound 
vertically  550  feet,  is  equal  to  the  amount  of  energy 


246  PRACTICAL  HYDRAULICS. 

necessary  to  raise  550  pounds  vertically  one  foot  in 
bight. 

This  amount  .of  energy  rendered  in  one  second  is 
termed  a  horse-power — that  is,  550  foot-pounds  ren- 
dered in  one  second,  is  the  value  of  a  horse  power  in 
mechanics. 

The  weight  of  a  cubic  foot  of  fresh  water  is  esti- 
mated in  practice  at  62.5  pounds. 

Ex.  118. — How  many  horse-power  will  10  cubic 
feet  of  water,  applied  to  an  overshot  water  wheel,  40 
feet  diameter,  render,  the  efficiency  of  the  wheel  being 
75  per  cent,  and  one  foot  being  allowed  for  clearance? 

Gal.— 40— 1=39  feet,  effective  head;  62.5X10X39 
X  .75H-550---33.24  horse-power.—  Ans. 

TABLE  30. 

Limiting  Velocities  in  Open  Streams.— Jackson's  Hydraulic 

Manual. 

Feet  per  Second. 

For  the  worst  or  most  sandy  soil 2.5 

For  sandy  soil  generally 2 . 75 

For  ordinary  loam 3 . 

For  firm  gravel  and  hard  soil 4 . 

For  brick  work,  ashlar,  or  rubble  in  cement  ....5.5  to  7.5 

For  hard,  sound,  stratified  rock 10 . 

For  very  hard  homogeneous  rock 14 .  or  15 . 

Limits  usual  for  canals 1 .  to  4 . 

Limits  for  irrigating  channels , 1 .  to  3 . 

Limits  for  sewers  and  brick  conduits 1 .  to  4 . 5 

Limits  for  self -cleans  ing  sewers  and  drainage  pipes  2.5  to  4.5 

Remark. — The  importance  of  the  data,  given  in 
Table  30,  will  be  seen  at  a  glance. 

Thus,  if  a  velocity  exceeding  2.5  feet  per  second  be 
given  a  stream  in  very  sandy  soil,  destruction  by  eros- 


I 

PRACTICAL   HYDRAULICS.  247 

ion  of  the  bed  of  the  canal  will  ensue;  while  on  the 
other  hand,  if  the  velocity  shall  not  exceed  one  foot 
per  second  at  first,  the  canal  will  be  liable  to  become 
choked  up  by  the  growth  of  vegetation. 

FLOOD-FLOW  OF  STREAMS. 

Various  formulse  have  been  devised  for  the  flow  of 
streams  in  times  of  floods.  These  empirical  formulae 
are,  at  best,  but  rough  approximates  to  the  true  flow. 

The  following  formula,  given  by  Fanning,  is  an  ex- 
pression for  the  "  recorded  flood  measurement  of 
American  streams"  in  New  England  and  the  Middle 
States: 

Q=200  M*,  (287) 

in  which  Q  denotes  the  number  of  cubic  feet  discharge 
per  second,  and  M  the  area  of  water  shed  in  square 
miles. 

As  California  is  more  mountainous  than  New  Eng- 
land and  the  Middle  States,  and  fully  as  subject  to 
heavy  downfall  of  rain  in  the  mountainous  regions,  it 
is  not  improbable  that  the  flood  discharge  here  will 
exceed  that  indicated  by  formula  (287).  Let  it  hence 
be  accepted  until  otherwise  determined. 

Ex.  119. — The  water-shed  of  the  main  Sacramento 
river  contains  twenty-four  thousand  seven  hundred 
and  eight  square  miles.  What  will  be  its  flood  dis- 
charge per  second? 

.— (24,708)1x200=915647  cubic  feet.— Ans. 


248  PEACTICAL   HYDRAULICS. 

TABLE  32. 

Miscellanies. 

1  cubic  foot  of  distilled  water  (U.  S.  standard),  ba- 
rometer 30  inches,  39.83°  Fahr.  =62.3793  Ibs. 

1  cubic  foot  of  distilled  water  (British  standard),  ba- 
rometer 30  inches,  62°  Fahr.---62.32l  Ibs, 

1  cubic  foot  of  distilled  water  (U.  S.  standard) = 
7.48052  gallons. 

1  cubic  inch  of  distilled  water  (U.  S.  standard)  = 
0.0361  Ibs. 

1  gallon  (U.  S.  standard) --231  cubic  inches= 
0.133681  cubic  feet— 8.3389  pounds  water. 

1  gallon,  imperial  (British  standard)  =277. 12,3  cu- 
bic inches=- 0.160372  cubic  feet=  10  Ibs.  water. 

1  gallon  (N.  Y.  statute  measure),  barometer  30 
inches,  39.83°  Fahr.=221.184  cubic  inches=8  Ibs. 
water. 

1  pound  avoirdupois=16  ounces=7000  grains  (U. 
S.  standard)=27.7015  cubic  inches. 

1  pound  Troy=l  pound  apothecary =12  ounces= 
5760  grains. 

1  ounce  avoirdupois=437.5  grains. 

1  ounce  Troy=l  ounce  apothecary =480  grains. 

1  chain=100  links=4  rods=66  feet— 792  inches. 
•80  chains=l  statute  mile=320  rods==1760  yards= 
5280  feet— 63, 360  inches. 

1  geographical,  nautical  or  sea-mile = 6,086. 5  feet  in 
longitude;  and  6,076.5  feet  in  latitude. 

1  league  (English) =3  nautical  miles. 

1  metre=3. 2808992=3.281  in  practice. 

1  square  metre=l  centiare=10.7643  square  feet. 

1  are=100  square  metres=1076.43  square  feet. 

1  cubic  metre=l  stare=  35.3166  cubic  feet. 

1  vara=2.75  feet. 


PRACTICAL   HYDRAULICS.  249 

1  legua  (Mexican) =5000  varas  linear=  13,750  feet 
=2.60417  miles. 

100  vara  lot=100  varas  square -=75625  square 
feet— 1.73611  acres. 

1  legua,  Mexican  (of  land)— 6.7817  square  miles— 
4340.27778  acres. 

1  acre=4  roods— 10  square  chains— 160  square 
rods— 43560  square  feet. 

1  section=l  square  mile— 640 'acres. 

1  township=36  sections=6  miles  square= 36  square 
miles. 

1  cubic  yard=27  cubic  feet=16,656  cubic  inches. 

1    hundredweight  (British)  =8  stone=112  pounds. 

1  ton  (long  ton),  commercial-— 20  hundredweight— 
2240  pounds. 

1  ton  (short  ton),  U.  S.=2000  pounds. 

1  quintal^  100  pounds. 

1  fathom=6  feet;  1  cable  length— 120  fathoms. 

1  point —-fg  of  an  inch. 

1  line=6  points^^  of  an  inch. 

12  inches=l  foot;  3  feet  =  l  yard. 

5J  yards=l  rod. 

1  foot  board  measure =1  foot  square  and  1  inch 
thick. 

12  feet  board  measure=l  cubic  foot. 

1  foot-pound-— work  required  to  raise  one  pound 
vertically  one  foot. 

1  second  foot-pound=work  required  to  raise  one 
pound  vertically  one  foot  in  one  second  of  time. 

1  minute  foot-pound—work  required  to  raise  one 
pound  vertically  one  foot  in  one  minute. 

1°  (one  degree),  centigrade =1.8°  (degrees),  Fah- 
renheit. 

1  barometric  inch— column  of  mercury,  with  one 
square  inch  base  and  one  inch  high. 


250  PKACTICAL    HYDRAULICS. 

Atmospheric  pressure  per  square  inch=14.7  pounds 
=30  barometric  inches  nearly,  at  39.83°  Fahr. 

1  ounce  Troy,  gold,  1000  fine=$20.67l8. 

1  ounce  Troy,  gold  coin,  U.  S.,  900  fine=$l8.6046. 

1  pound  avoirdupois,  gold  coin,  U.  S.,  900  fine= 
$271.375. 

1  ounce  Troy,  silver,  1000  fine-=$l.  29293. 
1  ounce  Troy,  silver,  U.  S.,  900  fine=$l.l63636. 
1  pound  avoirdupois,  silver  coin,  U.  S.,  900  fine= 
$16.96969. 

1  dollar,  U.  S.  gold  coin=23.22  grains  gold  +  2.  58 
grains  copper—  25.8  grains. 

1  dollar,  U.  S.  silver  coin=37l.25  grains  silver  + 
41.25  grains  copper=412.5  grains. 

1  pound  sterling—  1  so  vereign=l  13.001  grains  gold 
4-10.273  grains  copper  =;  12  3.  274  grains  weight,  fine- 
ness 22  carats=  916.6667. 

1  grain  gold,  1000  fine=-  $.0430663  mint  value. 

1  grain  silver,  1000  fine=-$.  0026936  mint  value. 

1  gramme  gold,  1000  fine—  $.6646142   mint  value. 

1  gramme  silver,  1000  fine—  $.  0415686  mint  value. 

1  cubic  foot  air^.  0806726  pounds  --=564.7082  gr's. 

1  pound  of  air  at  39.83°  =  12.387  cubic  feet  by  vol. 

1  cubic  foot  hydrogen^.  005042  pounds=35.2743 
grains. 

25  cubic  feet  of  sand=l  ton. 

18  cubic  feet  of  earth  =l'ton. 

17  cubic  feet  of  clay=-l  ton. 

13  cubic  feet  of  quartz,  unbroken  in  lode  =  l  ton. 

18  cubic  feet  of  gravel  or  earth,  before  digging  — 
27  cubic  feet  when  dug. 

20  cubic  feet  of  quartz,  broken  (of  ordinary  fineness 
coming  from  the  lode)  —  1  ton,  contract  measurement. 

1  horse-power  (H.  P.)=550  second  foot-pounds= 
33000  minute  foot-pounds. 


OF  THE 

WNIVER 


HOSKIN'S  NEW  HYDRAULIC  GIANT. 


Of  all  the  Hydraulic  Nozzles  or  Giants  made,  no  other  has  been  so  universally 
adopted  or  so  satisfactory  in  results.  It  is  no  exaggeration  to  say  that  it  is  now 
almost  exclusively  used  in  all  mining  countries,  and  its  great  superiority  every- 
where acknowledged,  so  that  the  Hoskin  Giant  has  come  to  be  regarded  as  the 
standard  for  this  class  of  appliances.  Several  improvements  have  recently  been 
made  which  greatly  increase  its  efficiency,  as  well  as  its  security  and  convenience. 

Horizontal  and  vertical  motions  are  now  made  with  only  one  joint,  and  this 
so  protected  as  to  be  durable  and  easily  kept  in  order.  The  Nozzle  Butt  is  at- 
tached to  the  pipe  in  such  a  way  as  to  counteract  the  downward  movement  com- 
mon to  this  class  of  machines  when  working  under  great  pressure.  The  pipe  is 
balanced  by  matching  the  notch  in  its  flange  with  corresponding  one  in  the 
flange  of  the  elbow. 

This  Machine  is  fully  secured  by  Letters  Patent  in  this, 
as  •well  as  other  mining  countries,  and  full  protection  guar- 
anteed to  all  purchasers. 

HOSKIN'S  &  PERKINS'  PATENT  DEFLECTORS. 

Special  attention  is  called  to  the  many  advantages  of  these  Deflectors,  and  to 
the  fact  that  they  are  adapted  to  no  other  Giants.  By  means  of  a  hand-lever  the 
stream  can  be  deflected  to  any  desired  angle  without  moving  the  body  of  the 
machine,  and  it  is  moved  so  easily  that  any  child  can  operate  it.  This  is  the 
most  valuable  invention  that  has  ever  been  applied  to  this  class  of  machines,  and 
gives  the  Hoskin  Machine  great  pre-eminence  over  all  others. 

ROMAN'S  SAFETY  PRESSURE  EQUALIZER. 

A  simple  and  most  effective  device  for  equalizing  the  strain  on  hydraulic 
pipes,  and  compensating  for*all  variations  of  pressure.  It  relieves  the  direct 
shock  upon  the  pipes,  and  prevents  the  collapse  or  disastrous  breaks  so  common 
from  this  cause.  It  is  an  indispensable  and  universally  recognized  necessity  for 
all  hydraulic  operations.  One  required  for  each  line  of  pipe. 
FOR  CIRCULAR  AND  PRICE  LIST.  Address 


PACIFIC    IRON    WORKS, 

127  First  Street,  1OO  N.  Clinton  St.,  145  Broadway, 

SAN  FRANCISCO.  CHICAGO.  NEW  YORK. 


JOSHUA  HENDY   MACHINE  WORKS, 

(INCORPORATED  SEPTEMBER  29,  1882.) 

Nos.  39  to  51  Fremont  St.,  San  Francisco,  Cal. 


MANUFACTURERS     OF- 


Hydraulic  Gravel  Elevators 

HYDRAULIC  GIANTS, 


HYDRAULIC    MINING    PIPE, 
QUARTZ  and  SAW-MILL  MACHINERY, 

NEW  and  SECOND-HAND  BOILERS,  ENGINES 

AND  MACHINERY  OF  EVERY  DESCRIPTION. 

AGENTS     FOR    THE    SALE    OF 

"Cummer"  Engines,  from  Cleveland,  Ohio,* 

Porter  Manufacturing  Co.'s  Engines  and  Boilers, 

"Baker"    Rotary    Pressure    Blowers, 
"Wilbraham"  Rotary  Piston  Pumps, 

"  Boggs  &  Clarke  "  Centrifugal  Pumps, 

The  Volker  &  Felthousen  Manuf'g  Co.'s 

Buffalo  Duplex  Steam  Pumps, 

P.  Blaisdell  &  Co.'s  Machinists'  Tools. 


1.64453 


